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Tài liệu Tổng quát giải tích hàm số Lớp 12

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  1. a a a aaa aa a a a
  2. a K a f K • K xx Kx x fx fx 12,∈ , 12 ( 2 ) f I • f I f'( x ) ≥ 0 x∈ I • f I f'( x ) ≤ 0 x∈ I aaa f a; b  (a; b ) c∈ ( a; b ) a fb( ) − fa( ) = fcba' ( ) ( − ) I a f I a I I a I • f'( x ) > 0 x∈ I f I • f'( x ) 0 (a; b ) f a; b  • f a; b  f'( x ) < 0 (a; b ) f a; b  • aa f I f'( x )≥ 0 ∀x ∈ I f'( x )≤ 0 ∀x ∈ I f'( x )= 0 a I f I
  3. O a a y= f( x )aa • D a • y'= f ' ( x ) • a x D f'( x ) = 0 f' ( x ) a • y'= f ' ( x ) x D • aaơa aa 1. y=− x3 − 3 x 2 + 24 x + 26 2. y= x3 − 3 x 2 + 2 3. yx=3 + 3 x 2 + 32 x + 1. y=− x3 − 3 x 2 + 24 x + 26 » a y'=− 3 x2 − 6 x + 24 x = − 4 y'0= ⇔ − 3 x2 − 6 x + 240 = ⇔  x =  2 a y ' x −∞ −4 2 +∞ y ' − 0 + 0 − y'> 0, x ∈−( 4;2 ) ⇒ y (−4;2 ) y'> 0, x ∈−∞−( ; 4) ,( 2; +∞⇒) y (−∞; − 4) ,( 2; +∞ ) a » a y'=− 3 x2 − 6 x + 24 x = − 4 y'0= ⇔ − 3 x2 − 6 x + 240 = ⇔  x =  2 x −∞ −4 2 +∞ y ' − 0 + 0 − +∞ y −∞ (−4;2 ) (−∞; − 4 ) (2; +∞ )
  4. 2. y= x3 − 3 x 2 + 2 » a y'3= x2 −= 6 x 3( xx − 2) x = 0 y'= 0 ⇔ 3( x x − 2) = 0 ⇔  x =  2 x −∞ 0 2 +∞ y ' 0 − 0 y (−∞ ;0) (2;+∞ ) (0;2) 3. yx=3 + 3 x 2 + 32 x + » 2 a fx'()()= 3 x2 = 633 x += x + 1 f'( x) = 0 ⇔ x =− 1 f'( x ) > 0 x ≠ − 1 » a (−∞; − 1  −1; +∞ ) a x −∞ −1 +∞ y ' + 0 + +∞ y 1 −∞ » a (−∞; − 1  −1; +∞ ) aa 1 1. y=− x4 + 2 x 2 − 1 4 2. y= x4 + 2 x 2 − 3 3. yx=4 − 6 x 2 + 8 x + 1 1 1. y=− x4 + 2 x 2 − 1 4 » a y'=− x3 + 4 x =− xx( 2 − 4 )
  5. x = 0 y'0= ⇔ − x x 2 − 40 = ⇔  () x = ±  2 x −∞ −2 0 2 +∞ y ' + 0 − 0 + 0 − y +∞ −∞ (−∞; − 2 ) (0;2 ) (−2;0 ) (2; +∞ ) 2. y= x4 + 2 x 2 − 3 » a y'4= x3 += 44 x xx( 2 + 1 ) x2 +1 > 0, ∀ x ∈ » y'= 0 ⇔ x = 0 x −∞ 0 +∞ y ' − + +∞ +∞ y (0; +∞ ) (−∞ ;0 ) 3. yx=4 − 6 x 2 + 8 x + 1 » a yxx'= 43 − 12 += 8 4( x − 1) 2 ( x + 2) x = − 2 y'= 0 ⇔ 4( x − 1)2 ( x + 2) = 0 ⇔  x =  1 x −∞ −2 1 +∞ y ' − 0 + 0 + y (− 2; +∞ ) (−∞ ; − 2) a x = 1 y = 0 a y ' y= ax4 + bx 3 + cx 2 ++ dx e
  6. ơ » aa 2x − 1 1. y = x + 1 x + 2 2. y = x − 1 −x2 +2 x − 1 3. y = x + 2 x2 +4 x + 3 4. y = x + 2 2x − 1 1. y = x + 1 (−∞−; 1) ∪( − 1; +∞ ) 3 a y'= >∀≠− 0, x 1 2 ()x + 1 (−∞; − 1 ) (−1; +∞ ) x + 2 2. y = x − 1 (−∞;1) ∪( 1; +∞ ) 3 a y'= - <∀≠ 0, x 1 2 ()x − 1 (−∞ ;1 ) (1; +∞ ) −x2 +2 x − 1 3. y = x + 2 (−∞−; 2) ∪( − 2; +∞ ) −x2 −4 x + 5 a y'= , ∀≠− x 2 2 ()x + 2 x = − 5 y '= 0 ⇔  x =  1 x −∞ −5 −2 1 +∞ y ' − 0 + + 0 − +∞ +∞ y −∞ −∞
  7. (−5; − 2 ) (−2;1 ) (−∞; − 5 ) (1; +∞ ) x2 +4 x + 3 4. y = x + 2 (−∞−; 2) ∪( − 2; +∞ ) x2 +4 x + 5 a y'= >∀≠− 0, x 2 2 ()x + 2 x −∞ −2 +∞ y ' + + +∞ +∞ y −∞ −∞ (−∞; − 2 ) (−2; +∞ ) ax+ b y= ( a . c ≠ 0) cx+ d a ax2 + bx + c y = aơ a' x+ b ' aơ » aa 1. y= | x2 − 2 x − 3 | 2. y= 3 x2 − x 3 1. y= | x2 − 2 x − 3 | »  2 xx−−2 3 khi x ≤−∪≥ 1 x 3 a y =  −++x2 x − 1 x 3 ⇒=y' ⇒=⇔= y '01 x −+2x 2 khi −<< 1 x 3  x = − 1 x = 3 x −∞ −1 1 3 +∞ y ' − 0 + 0 − 0 + y
  8. (− 1;1) (3;+∞ ) (−∞ ; − 1) (1;3) 2. y= 3 x2 − x 3 a (−∞ ;3] 3(2x− x 2 ) a y'= , ∀<≠ x 3, x 0 2 3 x2− x 3 ∀<xx3, ≠ 0:'0 y =⇔= x 2 x=0, x = 3 −∞ 0 2 3 +∞ x y ' − 0 − y (0;2) (−∞ ;0) (2;3) ơa f( x) = sin x (0;2 π ) (0;2 π ) a fx'( ) = cos xx , ∈ ( 0;2 π ) π3 π fxx'()()=∈ 0, 0;2π ⇔= xx , = 2 2 aa π 3π x 0 2π 2 2 f' ( x ) + 0 − 0 + f( x ) 1 0 0 −1 π  3π  π3 π  0;  ;2 π  ;  2  2  2 2 
  9. aa 2 1 3 2 x− 2 x 1. y= x − 3 x +− 8 x 2 2. y = 3 x − 1 aa 1. y= 2 x3 + 3 x 2 + 1 4 2 3. y=− x3 + 6 x 2 − 9 x − 2. y= x4 − 2 x 2 − 5 3 3 4. y= 2 x − x 2 2 1. y=4 − x 0;2  2. yx=3 +− xcos x − 4 » 3. y=cos2 x − 2 x + 3 » y=sin2 x + cos x π  π  a) 0;  ;π  3  3  b) m ∈( − 1;1 ) ơ sin2 x+ cos x = m 0; π  1 1. y= x3 − 3 x 2 +− 8 x 2 3 » a fx'( ) = x2 − 6 x + 8 fx'( ) =⇔= 0 x 2, x = 4 aa x −∞ 2 4 +∞ f' ( x ) + 0 − 0 + f( x ) +∞ −∞ (−∞ ;2 ) (4; +∞ ) (2;4 ) x2 − 2 x 2. y = x − 1 » \{ 1 }
  10. 2 x2 −2 x + 2 ()x −1 + 1 a f' x= = >≠ 0, x 1 () 2 2 ()x−1() x − 1 aa x −∞ 1 +∞ f' ( x ) + + +∞ +∞ f( x ) −∞ −∞ (−∞ ;1 ) (1; +∞ ) 1. y= 2 x3 + 3 x 2 + 1 » a fx'( ) = 6 x2 + 6 x fxx'( ) > 0, ∈−∞−( ; 1) ,( 0; +∞⇒) fx( ) (−∞; − 1 ) (0; +∞ ) fx'( ) 0, ∈−( 1;0) ,( 1; +∞⇒) fx( ) (−1;0 ) (1; +∞ ) fxx'( ) < 0, ∈−∞−( ; 1) ,( 0;1 ) ⇒ fx( ) (−∞; − 1 ) (0;1 ) a f'( x ) = 0 aa x=−1, x = 0, x = 1 4 2 3. y=− x3 + 6 x 2 − 9 x − 3 3 » 2 a fx'()()=− 4 x2 + 129 x −=− 23 x − 3 3 f'() x= 0 ⇔ x = f'( x ) < 0 x ≠ 2 2 3  3  a  −∞ ;   ;+∞  » 2  2 
  11. 4. y= 2 x − x 2 0;2  1 − x a f'() x= , x ∈ () 0;2 2x− x 2 fx'( ) > 0, x ∈( 0;1 ) ⇒ fx( ) (0;1 ) fx'( ) 0, x ∈( 0;1 ) ⇒ fx( ) 0;1  fx'( ) < 0, x ∈( 1;2 ) ⇒ fx( ) 1;2  2 1.y= 4 − x 0;2  −x 0;2  f'() x = < 0 x ∈ (0;2 ) 4 − x 2 0;2  2. yx=3 +− xcos x − 4 » » a fxx'( ) = 32 + 1 + sin x 3x2 ≥ 0 ∀ x ∈ »  f' x≥ 0, x ∈ » +x ≥ ∀∈ x » ( ) 1 sin 0 » 3. y=cos2 x − 2 x + 3 » » π a fx'( ) =− 2sin2( x +≤∀∈ 1) 0, x » fx'() =⇔ 0sin2 x =−⇔=−+ 1 x kkπ , ∈ » 4 π π  −+kπ; −++() k 1 π  , k ∈ » 4 4  » π  π  a) 0;  ;π  3  3  0; π  y'= sin x( 2 cos x − 1) , x ∈ ( 0; π )
  12. 1 π x∈(0;π ) ⇒ sin x > 0 ()()0;π : fx '=⇔ 0 cos x =⇔= x 2 3 π  π  • y' > 0, ∀∈ x  0;  0;  3  3  π  π  • y' < 0, ∀∈ x  ; π  ;π  3  3  b) m ∈( − 1;1 ) ơ sin2 x+ cos x = m 0; π  π  π  5 • x ∈ 0;  a y()0≤≤ yy  ⇔≤≤ 1 y ơ m ∈( − 1;1 ) 3  3  4 π  π  5 • x ∈ ;π  a yyy()π ≤≤  ⇔−≤≤1 y aa 3  3  4 5  π  ∀m ∈−()1;1 ⊂− 1;  c ∈ ;π  a y( c ) = 0 c aơ 4  3  2 π  sinx+ cos x = m ;π  ơ 3  ơ 0; π  » • f( x )ơ » f'( x) ≥ 0, ∀ x ∈ » • f( x )ơ » f'( x) ≤ 0, ∀ x ∈ » m a » 1 yfx=() =− x3 +2 x 2 +() 21 m + xm −+ 32 3 » a y'=− x2 + 421 x + m + ∆' = 2m + 5 ∆ ' m −∞ 5 +∞ − 2 ∆ ' − 0 + 5 2 • m = − y'=−() x − 2 ≤ 0 x∈», y ' = 0 x = 2 2 » 5 • m < − y'< 0, ∀ x ∈ » » 2
  13. 5 • m > − y = a xx x x x x ' 0 12, ( 1 0 x ∈ » y » 2 • a = 2 y'=() x + 2 a y'=⇔ 0 x =− 2, yx ' > 0, ≠− 2 y a » (−∞−; 2v à  −+∞ 2; ) y • ơ a = − 2 y » a a • 2 y '= 0 a x1, x 2 x1 (x1; x 2 ) (−∞ ;x1 ) (x2;+∞ ) 2 2 y » −2 ≤a ≤ 2 m y= xm + cos x » » a y'= 1 − m sin x » ⇔≥∀∈yx' 0,» ⇔− 1 mxx sin ≥∀∈⇔ 0, » mxx sin ≤∀∈ 1, » (1) m = 0 (1) 1 1 m > 0 (1)⇔ sinx ≤ ∀∈ x» ⇔≤ 1 ⇔<≤ 0 m 1 m m 1 1 m < 0 (1)⇔ sinxxR ≥ ∀∈ ⇔−≥ 1 ⇔−≤ 1 m < 0 m m −1 ≤m ≤ 1 »⇔y' ≥ 0 ∀∈ x »
  14. 1−m ≥ 0 ⇔min'min{1ymm = −+≥⇔ ;1 }0 ⇔−≤≤ 1 m 1 +m ≥ 1 0 y= fxm( , ) »⇔≥∀∈⇔y'0 x » min y '0 ≥ x∈» y= fxm( , ) »⇔≤∀∈⇔y'0 x » max y '0 ≤ x∈» y' = ax2 + bx + c a= b = 0  c ≥  0 y'≥ 0 ∀ x ∈» ⇔  a > 0   ∆ ≤ 0  a= b = 0  c ≤  0 y'≤ 0 ∀ x ∈» ⇔  a < 0   ∆ ≤ 0  » » m a » x 3 yfxm=() =+( 2) −+ ( mxm 2)2 +−+−() 8 xm 2 1 3 m a » 1 ayfx. =() = m2 − 1 x 3 ++() m 135 x 2 ++ x 3 () (m−1) x2 + 2 x + 1 by. = fx() = x + 1 a m a 2 m −2x +( m + 231) x − m + a. y= x + 2 + b. y = x − 1 x − 1 x 3 yfxm=() =+( 2) −+ ( mxm 2)2 +−+−() 8 xm 2 1 3 » a y'=+ ( m 2) x2 − 2( m + 2) xm +− 8
  15. m = − 2 y'=− 10 ≤ 0, ∀∈ x » ⇒ » m ≠ − 2 a y'=+ ( m 2) x2 − 2( m + 2) xm +− 8 ∆' = 10(m + 2) ∆ ' m −∞ −2 +∞ ∆ ' − 0 + • m − y = a xx x x 2 ' 0 12, ( 1 ∀∈ x» ⇒ a =− 1 • a2 −1 ≠ 0 ⇔ a ≠± 1 ∆ ' a −∞ −1 1 2 +∞ ∆ ' − 0 + 0 − • a 2 y '> 0 x ∈ » y » 2 • a = 2 y'= 3() x + 1 a y'=⇔ 0 x =− 1, yx ' > 0, ≠− 1 y » a (−∞−; 1va `  −+∞ 1; ) y • −<1a < 2, a ≠ 1 y '= 0 a x1, x 2 x1< x 2 a a (x1; x 2 ) (−∞ ;x1 ) (x2;+∞ ) −<1 < 2, ≠ 1 y » a<−1 ∨ a ≥ 2 (m−1) x2 + 2 x + 1 by. = fx() = x + 1 D =» \{ − 1 } (mx−1) 2 + 2( mx − 1) + 1 gx( ) ay ' = = , 2 2 ()x+1() x + 1 gx( ) =( m −1) x2 + 2( m − 1) x + 1, x ≠− 1
  16. ay 'ag( x ) y (−∞; − 1 ) (−1; +∞ ) g( x) ≥0, ∀ x ≠− 1 ( 1 ) • m−=⇔10 mgx =⇒ 1( ) =>∀≠−⇒ 10, x 1 ma = 1 ( ) • m−≠1 0 ⇔ m ≠ 1 ơ 1 0; ∀ x ≠ 1 (−∞ ;1 ) (1; +∞ ) 2 m ()x−1 − m • m > 0 y'1=− = ,1 x ≠ y'= 0 ⇔ x = 1 ± m a 2 2 ()x−1() x − 1 (1− m ;1 ) (1;1 + m ) a m ≤ 0 a a m 1)a (−∞; − 1 ) a m 2 ) a (2; +∞ ) a3 ) a m a m 4 )a (0;1 ) (1;2 ) 2 a x x m 5 ) 1< 2 aaơ ()x−1 − m = 0 a5.1 ) x1= 2 x 2 a5.3 ) x1+3 x 2 < m + 5 a5.2 ) x1< 3 x 2 a5.4 ) x1−5 x 2 ≥ m − 12 2 −2x +( m + 231) x − m + 1− 2 m by. = =−++ 2 xm x−1 x − 1 2m − 1 ⇒y ' =− 2 + 2 ()x − 1 1 • m ≤⇒< y' 0, x ≠ 1 (−∞;1) va `( 1; +∞ ) 2
  17. 1 • m > ơ y '= 0 a x<1 < x ⇒ 2 1 2 x v x a ( 1;1) à( 1; 2 ) oa » y= fxm( , ) ∀x ∈ I ⇔≥∀∈⇔y' 0 xI min y ' ≥ 0 x∈ I y= fxm( , ) ∀∈⇔xIy'0 ≤∀∈⇔ xI max'0 y ≤ x∈ I m a mx + 4 y= f x = −∞ ;1 () x+ m ( ) yx=+33 x 2 +( m + 1) xm + 4 (−1;1 ) mx + 4 y= f() x = (−∞ ;1 ) x+ m D=» \ { − m } m2 − 4 ay'= , x ≠ − m 2 ()x+ m y'< 0, ∀ x ∈( −∞ ;1 ) (−∞ ;1 )  −m ∉ −∞  ();1 2 m −4 < 0 −<2m < 2  −<< 2 m 2 ⇔ ⇔  ⇔  ⇔−<≤−2m 1 −m ∉ −∞ −≥m1 m ≤− 1  ();1   −2 <m ≤− 1 yx=+33 x 2 +( m + 1) xm + 4 (−1;1 ) » a fx'( ) = 36 x2 + xm ++ 1 (−1;1 ) f'( x) ≤ 0, ∀ x ∈−( 1;1 ) a mxx≤−(32 + 61, +) ∀∈− x( 1;1) ⇔ m ≤ min gx( ) ( 1 ) x∈() − 1;1 gx( ) =−(3 xx2 + 6 + 1,) ∀∈− x ( 1;1 ) ⇒gx'( ) =− 6 x −<∀∈− 6 0, x( 1;1 ) ⇒ gx( ) (−1;1 ) limgx( ) =− 2,lim gx( ) =− 10 x→−1+ x → 1 −
  18. x −1 1 g' ( x ) − g( x ) −2 −10 m ≤ − 10 f''( x) = 6 x + 6 aơ f''( x ) = 0 x = −1 gx( ) =6 x2 − 4 x (1; +∞ ) a gx'( ) = 12 x − 40, > ∀ x > 1 ⇔ gx( ) (1; +∞ ) limgx= lim6 xx2 −= 4 2,lim gx =+∞ +( ) + ( ) ( ) x→1 x → 1 x →+∞ x 1 +∞ g' ( x ) + g( x ) +∞
  19. 2 aa 2≥−m ⇔ m ≥− 2 2. y= fx( ) = mx3 −++− x 2 3 xm 2 (−3;0 ) (−3;0 ) a y'3= mx2 − 2 x + 3 (−3;0 ) y'≥ 0, ∀ x ∈−( 3;0 ) 2x + 3 a 3mx2 − 2 x +≥∀∈− 3 0, x() 3;0 ⇔≥ m , ∀∈− x () 3;0 3x 2 2x + 3 g() x = (−3;0 ) a 3x 2 −6x2 + 18 x gx'() = <∀∈− 0, x()() 3;0 ⇒ gx 9x 4 1 (−3;0 ) limgx() =− ,lim gx() =−∞ x→−3+9 x → 0 − x −3 0 g' ( x ) − 1 g( x ) − 9 −∞ 1 aa m ≥ − 9 1 3. yfx=() = mx3 +−+−+2()() m 1 x 2 m 1 xm 3 (2; +∞ ) (2; +∞ ) a y'= mx2 + 41( m −) x +− m 1 (2; +∞ ) y'≥∀∈ 0, x( 2; +∞⇔) mxmxm2 + 4( − 1) + −≥∀∈ 1 0, x ( 2; +∞ ) 4x + 1 ⇔xxmxx2 ++4 1 ≥+∀∈+∞⇔≥ 4 1, 2; m , ∀∈+∞ x 2; () () 2 () x+4 x + 1 4x + 1 g() x=, x ∈() 2; +∞ x2 +4 x + 1
  20. −2x( 2 x + 1 ) ⇒gx' = <∀∈+∞⇒ 0, x 2; gx 2; +∞ () 2 ()() ( ) ()x2 +4 x + 1 9 limgx() = ,lim gx() = 0 x →2+ 13 x →+∞ x 2 +∞ g' ( x ) − 9 g( x ) 13 0 9 m ≥ 13 a m y= x3 +3 x 2 + mx + m 1 » a y'= 3 x2 + 6 xm + ∆' = 9 − 3 m • m ≥ 3 y'≥ 0, ∀ x ∈ » » m ≥ 3 • m < y 3 '= 0 a xx12, ( x 1< x 2 ) x; x  l= x − x 1 2  2 1 m a x+ x =−2, xx = 1 2 12 3 1⇔l = 1 2 2 4 9 xx xxxx m m ⇔−()()21 =⇔+1 12 − 414 12 =⇔− =⇔= 1 3 4 a m y= x3 +3 x 2 + mx + m 1 a l= x2 − x 1 ≥ 1?. aam a a. yxmx=−−3 2(2 m 2 −++ 7 m 7) x 2( m − 123)( m − ) (2; +∞ )
  21. mx2 +( m +1) x − 1 b. y = (1; +∞ ) 2x− m 3 2 2 m yx=−+( m 1) x − (2 m −++ 3 m 2) xmm (2 − 1) 2; +∞ ) mx2 +6 x − 2 m y = [1;+∞ ) x + 2 1 m y= mx3 −−( m 1) x 2 + 3( m −+ 2) x 1 (2;+∞ ) 3 a. » a y'32= x2 − mx −( 2 m 2 −+= 77 m) gx( ) (2; +∞ ) y'≥ 0, ∀ x ∈( 2; +∞ ) gx( ) =−32 x2 mx −( 2 m 2 −+ 77 m ) x ∈(2; +∞ ) gx'( ) = 6 x − 2 m g( x ) (2; +∞ ) g(2) ≥⇔ 0 3.22 − 2.2 mmm −( 2 2 −+≥ 7 7) 0 5 ⇔−2m2 + 3 m +≥⇔−≤ 50 1 m ≤ 2 a m g'() x= 0 ⇔ x = 3 m • ≤2 ⇔m ≤ 6 g( x) ≥0, x ∈( 2; +∞ ) 3 5 ⇔mingx() ≥⇔− 0 2 mm2 + 3 +≥⇔−≤≤ 50 1 m x∈()2; +∞ 2 m • >2 ⇔m > 6 aa 3 b. m  D = » \   2  x − 1 1 • m = 0 ay= ⇒= y' >∀≠ 0, x 0 2x 2x 2 (−∞;0) v à ( 0; +∞ ) (1; +∞ )
  22. m= 0 ( a ) 2mx2− 2 m 2 x − m 2 −+ m 2 g( x ) • m ≠ 0 ay ' = = , 2 2 ()2xm−() 2 xm − gx( ) =2 mx2 − 2 mx 2 −−+ m 2 m 2 (1; +∞ ) 2m > 0   m > 0 m  ∉+∞()1; ⇔≤  m 2 ⇔ ∀∈ 70 m » f( x ) m+−∆1' m ++∆ 1' a x=; x = 13 2 3 x≤ x x 0 ∀ x ∈ » f( x ) a x, x 2 1 2
  23. f( x )≤ 0 ⇔x ∈ ( x1 ; x 2 ) (*) a 7 • m f( x )= 0 a 2 −+2mmm 42 − 14 −− 2 mmm 4 2 − 14 x; x = 1m 2 m 7 x≤ x m ⇒x < x ⇒ fx( ) ≤ 0 ⇔  1 2 1 2 x≥ x  2 fx≤∀∈+∞⇔ x x ≤⇔− mmm ≥2 − ()0 [1;)2 1 3 4 14 m < 0 14 ⇔ ⇔≤−m m2 + m ≥ 5 5 14 0 −14 (*)⇔≤m = gxx ( ) ∀∈+∞⇔≤ [1; ) m min gx ( ) x2 + 4 x x ≥1 14 14 a mingxg ( )= (1) =− ⇒ m ≤− x ≥1 5 5 a y'= mx2 − 2( m −+ 1) x 3( m − 2) ∀x ∈(2; +∞ ) • m = 0 y'= 2 x − 6 y '≥ 0 x ≥ 3 • m ≠ 0 ∆=−' 2m2 + 4 m + 1 2 ơa m ≥ 3 ∀x ∈(2; +∞ ) ⇔y' ≥ 0 ∀∈ x ( 2; +∞ ) ⇔mx2 −2( m −+ 1) x 3( m −≥ 2) 0 ∀x ∈(2; +∞ ) 6− 2 x ⇔≥m = gxx( ) ∀∈+∞() 2; x2 −2 x + 3 g( x ) a 2; +∞ ) 2(x2 − 6 x + 3) a g'( x ) = ∀x ∈2; +∞ ) (x2− 2 x + 3) 2 ⇒g'( x ) =⇔=+ 0 x 3 6 ( vi ` x ≥ 2) limg () x = 0 x →+∞ 2 a maxgx ( )= g (2) = x ≥2 3 2 ⇒≥mgx( ) ∀∈+∞⇔≥ x [2; ) mmaxgx ( ) = x ≥2 3
  24. a • a fx( ) ≥ Mx, ∈ ( ab ; ) • y= fxx( ), ∈ ( ab ; ) • a (a; b ) • a π  sinx+ t ax n > 2 x , ∀∈ x  0;  2  π  fx( ) =sin x + tn ax − 2 x a 0;  2  1 1 π  a fxx'= cos + −> 2cos2 x + −>∀∈ 20, x 0; () 2 2   cosx cos x 2  π  π  ⇒ f( x ) 0;  f( x) > f (0) , ∀x ∈ 0;  2  2  π  a sinx+ t ax n > 2 x , ∀∈ x  0;  2  π  1. sinx≤ x , ∀ x ∈  0;  2  x 3 π 2. sinx>− x , ∀∈ x (0; ) 3! 2 x2 x 4 π 3. cosx cosx , ∀ x ∈ (0; ) x  2 π  1. sinx≤ x , ∀ x ∈  0;  2  π  fx( )= sin xx − x ∈ 0;  2  π  π  a fx'( )= cos x −≤∀∈ 1 0 , x  0;  ⇒ f( x ) 0;  2  2  π  a fxf( )≤ (0) =⇔ 0 sin xxx ≤∀∈  0;  2 
  25. x 3 π 2. sinx>− x , ∀∈ x (0; ) 3! 2 x 3 π  fx( )= sin xx − + a x ∈  0;  6 2  x 2 π  a fx'()cos= x −+ 1 ⇒ fx "() =− sin xx +≥∀∈ 0 x  0;  2 2  π  π ⇒fxf'( ) ≥ '(0) =∀∈ 0 x 0; ⇒ fxf ( ) ≥ (0) =∀∈ 0 x  0;  2  2 x 3 π  ⇒sinx >− x , ∀∈ x  0;  3! 2  x2 x 4 π 3. cosx cosx , ∀ x ∈ (0; ) x  2 x 3 π  a sinx>− x , ∀∈ x  0;  6 2  3 3 sinxx2 sin x   x 2  xxx 246 ⇒ >−⇒1  >− 1  =−+− 1 x6 x  6  2 12 216     3 sin x  xxx2 4 4 x 2 ⇒  >−++1 (1 − ) x  2 24 24 9 3 π x2 sin x xx2 4 x ∈0; ⇒−>⇒ 1 0  >−+ 1 2 9 x 224 x2 x 4 π  1− + > cosx , ∀∈ x  0;  224 2  3 sin x π a >cosx , ∀ x ∈  0; x 2 sin x π a 0< sinx <⇒< x 0 < 1 ∀∈ x (0; ) x 2
  26. α 3 sinx  sin x   ≥  ∀α ≤ 3 aa x  x  α sin x  π ∀α ≤ 3 a   ≥cosx ∀ x ∈ (0; ) x  2 1 1 4 π  cosx , ∀ x ∈  0; x 2 3 3 π  π ⇒−xxxcos + sin >∀∈ 0 , x 0; ⇒ fx '( ) >∀∈ 0 , x  0; 2  2 π4  π ⇒fxf() ≤  =− 1 , ∀∈ x  0;  2π 2  2 1 1 4 π  2 2 2 1 sinx+ t a n x a 22.sinx+ 2 tn ax ≥ 2.2 2sin xax .2 tn = 2.2 2 1 3 x sinx+ t a n x 1 3 π  a 22≥⇔ 2 2 sinx + tn ax ≥ x ∀x ∈ 0;  2 2 2  1 3 x π  fx() =sin x + t ax n − a 0;  2 2 2  1 3 2cos3x− 3cos 2 x + 1 a f, () x=cos x + −= 2.cos2x2 2 cos 2 x (cosx− 1)2 (2 cos x + 1) π = ≥∀∈0 , x [0; ) 2 cos 2 x 2 π 1 3 π ⇒ f( x ) [0; ) ⇒fxf( ) ≥ (0) =⇒ 0 sin x + tan xx ≥ ∀x ∈ [0; ) 2 2 2 2
  27. a n > 1 nn n n n1+ +− n 1 0 x444+++ y z xyzxyz( ++≥ )( xyx 22 ++ y )( yzy 2222 ++ z )( zxz + x ) xz y xyz 1 x≥ y ≥ z ≥ 0 + + ≥ + + z yx yz x x≥ y ≥ z ≥ 0 xzy xyz  f( x ) =++− ++  zyx yzx  11y z 11 a fx'()(=−− )( − )( =− yz )( − )0, ≥∀≥ x 0 zyx2 x 2 yz x 2 ⇒ f( x ) ∀x ≥ 0 ⇒fx() ≥ fy () =⇒ 0 2. x, y , z > 0 x444+++ y z xyzxyz( ++≥ )( xyx 22 ++ y )( yzy 2222 ++ z )( zxz + x ) a x≥ y ≥ z > 0 fx()=+++ x444 y z xyzxyz ( ++− )()()() xyx 22 +− y yzy 2222 +− z zxz + x a fx'()4= x3 − 3() xyz 2 ++ xyzyzxyz + ( ++−+ )( y3 z 3 ) ⇒fx"( ) = 12 x2 − 6 xyz ( ++ ) 2 yz
  28. ⇒f"( x ) > 0 x≥ y ≥ z ⇒fx'() ≥ fy '() =−= zyz2 3 zyz 2 ( −≥ )0 f( x ) ⇒fx()() ≥ fy =− z4 2 zyyz 3 + 222 = zzy ()0 −≥⇒ 2 a b c 3 1. a, b , c > 0 + + ≥ ab+ bc + ca + 2 2. 0 0 + + ≥ ab+ bc + ca + 2 b c a 1 1 13 x=, y = , z =⇒ xyz = 1 ⇔ + + ≥ a b c 1+x 1 + y 1 + z 2 1 1 2 2 z z≤1 ⇒ xy ≥ 1 + ≥ = 1+x 1 + y 1+xy 1 + z 1112z 12 t 1 ⇒++≥ +=+ = f( t ) t= z ≤ 1 111+++xyz1+z 11 ++ zt 1 + t 2 2 2t 2(1− t ) a f'() t = − ≤ ≤ 0 (1+t )2 (1 + t 22 ) (1 + t 22 ) 3 ⇒ftf( ) ≥ (1) = , ∀≤ t 1 ⇒ 2 2a 2 b 2 c () ca− 2 2. 0 <a ≤ b ≤ c + + ≤+3 bccaab+++ aca( + ) b c =α, =x ,1 ≤≤ α x a a 222α x x2 + x + 4 + + ≤ α+++x1 x 1 α x + 1 x+1 2 x ( x + 1) ⇔++≥x2 x 1 (2 ++ 2α ) α+x 1 + α x+1 2 x ( x + 1) f( x )=++− x2 x 1 (2 ++ 2α ), 1 ≤≤ α x α+x 1 + α 2(2x + 1)α − 1 a f'()21 x=+− x − 2 α + 1 (x + α ) 2
  29. 2+1x 2  f'( x )=− (α 1) −  ≥ 0, 1 ≤≤ α x 2 α+1 (x + α )  1 f( x ) f() x≥ f ()α = α2 − 3 α +− 3 α 1 1 1 f '()23αα=−+=++−≥ αα 33 3 αα −= 30 α2 α 2 α 2 ⇒fx( ) ≥ f (α ) ≥ f (1) =⇒ 0 fx( ) =2sin xaxx + tn − 3 π  a)a 0;  2  π  b) 2sinx+ tn ax > 3 x x ∈ 0;  2  π  a) ta n x> x x ∈ 0;  2  x 3 π  b) ta n x> x + x ∈ 0;  3 2  4 π  fx() = x − t ax n x ∈ 0;  π 4  π  a)a 0;  4  4 π  b)a x≥ t a n x x ∈ 0;  π 4  a a) sin x 0 sin x> x x 1 − x ≠ 0 2 x 3 x 3 c) sin x> x − x > 0 sin x 2 x x ∈ 0;  2  ae. x ≥ 1 + x , ∀ x
  30. x 2 be. x ≥++ 1 x , ∀≥ x 0 2 1 ln(1+x ) ≥− x x2 ∀≥ x 0 2 a a ∀x ≥ 0 ln(1+x ) ≥ x − ax 2 a a ax ≥+1 x ∀≥ x 0 b a a1  b 1  a≥ b > 0 2+  ≤ 2 +  2a  2 b  y x (23xx+) > y 0 x+ b b x+ a   a xab, ,> 0, a ≠ b  >  x+ b   b x>ln1( + x) , ∀> x 0 x ∈(4 ; + ∞ ) a 2x > x 2 (x; y )a (xy− )xy+ = ( xy + ) xy − π  a)a 0;  2  π  fx( ) =2sin x + tan xx − 3 a 0;  2  1 2cos3x+ 1 − 3cos 2 x f'() x= 2 cos x + −= 3 cos2x cos 2 x 2 ()()1− cosx 2 cos x + 1 π  f'() x= >∀∈ 0, x  0;  cos 2 x 2  π  fx( ) =2sin x + tan xx − 3 a 0;  2  π  b) 2sinx+ tan x > 3 x x ∈ 0;  2  π  π  fx( ) =2sin x + tan xx − 3 a 0;  fxf()()≥0 =∀∈ 0, x  0;  2  2  π  π  2sinx+ tn ax − 3 x > 0 x ∈ 0;  a 2sinx+ tn ax > 3 x x ∈ 0;  2  2 
  31. π  a) fx( ) =t axx n − a 0;  2  π  fx( ) =t axx n − a 0;  2  1 π  fx'= −= 1 t axx n2 >∀∈ 0, 0; () 2   cos x 2  π  π  fx( ) =t axx n − a 0;  fxf()()>0 =∀∈ 0, x  0;  a 2  2  tan x> x x 3 π  b) ta n x> x + x ∈ 0;  3 2  x 3 π  gx() =t ax n − x − a 0;  3 2  x 3 π  gx() =t ax n − x − a 0;  3 2  1 gx'() = −−= 1 xaxx2 tn 2 − 2 cos 2 x π  gx'()()()= t axxaxx n − t n +>∀∈ 0, x  0;  a) 2  x 3 π  π  gx() =t ax n − x − a 0;  gxg()()>0 =∀∈ 0, x  0;  3 2  2  x 3 π  a ta n x> x + x ∈ 0;  3 2  π  a)a 0;  4  4 π  fx() = x − t ax n 0;  π 4  4 1 4 − π2  π  fx'() =− = − tn,0;, axx ∀∈   πcos 2 x π 4  4 − π fx'() = 0 ⇔ tn ax = π 4 − π π π  4 − π 0 ∈ 0, x( 0; c ) ⇒ f( x ) x∈ 0; c  π  π  • fx'() <∈ 0, x c ;  ⇒ f( x ) x∈  c ;  4  4 
  32. π  4 4 π  b) 0≤≤∀∈fxfcx()() ;0;  ⇒− xax tn0 ≥ hayxax ≥ tn x ∈ 0;  4  π π 4  a) sin x 0 π  fx( ) = x − sin x a 0;  2  2 x π  π  fx'() =− 1 cos x = 2 sin >∀∈ 0, x  0;  a 0;  a 2 2  2  π  π  π fxf()()>0 =∀∈ 0, x  0;  x−sin x >∀∈ 0, x 0; hayx > sin xx , ∀∈  0; 2  2  2 x 2 b) cosx > 1 − x ≠ 0 2 x 2 f() x=cos x − 1 + a 0; +∞ ) fx'( ) = x − sin x > 0 2  x > 0 a f( x )a 0; +∞ ) a fxf( ) >(0) =∀> 0, x 0 x 2 cosx−+ 1 > 0, ∀> x 0 2 2 ()−x x 2 x ∀ ∀ 1 − x ≠ 0 2 x 3 c) fxx() = − − sin x f'( x) f(0) khi x  ()()0 0 π  d) sinx+ t ax n > 2 x x ∈ 0;  2  π  fx( ) =sin x + tan xx − 2 a 0;  2  12 1 π  fxx'() = cos + −> 2 cos x + −>∀∈ 2 0, x  0;  a cos2x cos 2 x 2  π  π  0; a fxf()()>0 =∀∈ 0, x  0;  2  2  ae. x ≥ 1 + x , ∀ x fx( )= ex − x − 1 » a fxe'()=x −⇒ 1 fx '()0 =⇔= x 0
  33. a fxf( )≥ (0) = 0 ∀ x x 2 be. x ≥++ 1 x , ∀≥ x 0 2 x 2 fx( )= ex −− 1 x − a 0; +∞ ) 2  a fxe'()=x −− 1 x ≥ 0 ∀ x ⇒fxf( ) ≥ (0) =∀≥ 0 x 0 1 fx()= ln(1 + xx ) −+ x 2 a 0; +∞ ) 2  1 x 2 a fx'()= −+= 1 x ≥∀≥ 0, x 0 1+x x + 1 ⇒fxf( ) ≥ (0) = 0 ∀≥⇒ x 0 (1) a a ∀x ≥ 0 ln(1+x ) ≥ x − ax 2 (1) ∀x ≥ 0 ⇒ ∀x > 0 ln(1+x ) − x ⇔ ≥−∀>a x 0 x 2 ln(1)+x − x 1 1 x → 0+ a ⇒− ≥−a ⇔ a ≥ x 2 2 2 1 2 2 x− x ≥− xax ∀≥ x 0 2 1 ln(1+x ) ≥− x x2 ∀≥ x 0 2 a ln(1+x ) ≥− x ax2 ∀≥ x 0 1 a = 2 fx()= ax − x −≥ 10 x ≥ 0 a f( x ) [0;+∞ ) fx'()= ax ln a − 1 • 0 f x aơ x '() 00 log(ln)a 0 '( ) a x min()fx= fx ( ) 0 x ≥0 0 1 ⇒f() x ≥ 0 ∀ x ≥ 0 ⇔ f() x≥⇔ 0 + log(ln)1 a −≥ 0 0 ln a a
  34. 1 ln(lna ) ⇔ + −≥1 0 ⇔+1 ln(lna ) − ln a ≥ 0 lna ln a eln a ⇔ln ≥⇔ 0eaa ln ≥⇔ eaa ln −≥ 0 a ga()= e ln a − a 1 ∀∈ 1 0 aegage (1;) ⇒ () 0, t 0 (0; +∞ ) t2 (4t + 1 ) a≥ b >0 ⇒ (2) ơ a x>ln1( + x) , ∀> x 0 fxx( ) = −ln( 1 + x ) a 0; +∞ ) 1 f'() x = 1 − > 0 x > 0 f( x )a 0; +∞ ) ơa x + 1  f( x) > f (0) = 0 x > 0 a x>ln1( + x) , ∀> x 0 f( x )= 2 x − x 2 (4 ;+ ∞ ) a fx′()=− 2ln2x 2, xfx ′′ () = 2ln2 x 2 − 2 1 1 ln2>⇒ ln22 >⇒ 2ln2x 2 >⇒ 4 2ln22 x 2 −>∀> 0,x 4 ⇒f′′ () x > 0, ∀> x 4 2 4 fx′( )> f ′ (4) , ∀ x > 4 f′(4)= 24 ln2 −>⇒ 8 0 fxx ′ ( ) >∀> 0, 4 fxf()> (4)0, =∀>⇒ x 4 2x > xx2 , ∀> 4
  35. ln x a 2 h( x ) = (4 ;+ ∞ ) x a ()0xy−x+ y >⇒−≠ xy 0 x− y 0 −()x − y (2k ) a 2 a (xy− )xy+ = ( xy + ) xy − a (xy+ )ln( xy −=− ) ( xy )ln( xy + ) ln(xy− ) ln( xy + ) ⇔ = ()* xy− xy + ln t f( t ) = 0; +∞ t ( ) 1− ln t a f′( t )= , t ∈+∞() 0; t 2 ln t ft′( )= 0 ⇔ te = f( t ) = ơ (0 ;e ) (e ;+ ∞ ) t ơ (*) fx(− y )( = fx + y ) 0 x 2 (* *) ⇒x + y ≤ 4 ( b ) x+ y =4  x = 3 a b a ⇔  ( ) ( ) x− y = y = 2  1 ư ư y= f( x ) ơ D D aơ f( x) = k ơ fx( ) = fy( ) x= y • y= f( x ) ơ D D y= g( x ) ơ D D aơ fx( ) = gx( ) ơ • y= f( x ) n D ơ f(k ) ( x )= 0 m ơ f(k − 1) ( x )= 0 m + 1
  36. ơ 1. 3x (2+ 9 x2 ++ 3) (4 x + 2)( 1 +++= xx 2 1) 0 3 2. xxx3−4 2 −+= 56 7 xx 2 +− 94 1. 3(2x+ 9 x2 ++ 3)(4 x + 2)(1 +++= xx 2 1) 0(1) ơ (1)⇔−() 3x (2 +− (3) x2 +=+ 3) (2 x 1)(2 + (2 x ++ 1)2 3)(2) u=−3, xv = 2 x + 1,, uv > 0 ơ (1)⇔u (2 + u2 +=+ 3) v (2 v 2 + 3) (3) ft()2= t + t4 + 3 t 2 (0; +∞ ) 2t3 + 3 t a ft'( )=+ 2 > 0, ∀>⇒ tft 0 () (0; +∞ ) t4+ 3 t 2 1 ơ (3)⇔fufv ( ) = ( ) ⇔=⇔−= uv 3 xx 2 +⇔=− 1 x 5 1 x = − aơ 5 3 2. xxx3−4 2 −+= 56 7 xx 2 +− 94 3 2 3 x−4 x − 5 x += 6 y y=7 x2 + 9 x − 4 ơ ⇔  x2+ x − = y 3 7 9 4 3 2  3 2 x−4 x − 5 x += 6 y  x−4 x − 5 x += 6 y ⇔3 3 2 ⇔  3 ()I yyx+= +3 x + 4 x + 2 yyx3 += + ++ x   ()()1 1 * (*) fy( ) = fx( + 1) ( a ) ft( ) = t3 + tt, ∈ » ftt'( ) = 32 +> 1 0, ∀∈ t » » (a) ⇔ y = x + 1 3 2 x−4 x − 5 x += 6 y x3−4 x 2 − 6 x += 5 0 ( ) I ⇔ ⇔  () y= x + y= x +  1  1 −+1 5 −− 1 5  ơ (* * ) a S = 5, ,  2 2  ơ 2x2 x − 211 =
  37. 2 y=2 x x − 2 a 2; +∞ ) x(5 x − 8 ) a y'= >∀∈+∞ 0, x 2; limy= lim2 x2 x − 2 = +∞ () x→+∞ x →+∞ ( ) x − 2 x 2 +∞ y ' + y +∞ 0 aaay=2 x2 x − 2 y = 11 ơ 2x2 x − 211 = 2 y= fx( ) =2 xx −− 211 a 2; +∞ ) a f(2) = − 11, f ( 3) = 7 ff(2.3) ( ) =− 770 ∀∈+∞⇒ 0, x()() 2; fx (2;3 ) x − 2 ơ (2;3 ) ơa 51x−+ x + 34 ≥ 1 x ≥ 5 1  fx()= 51 x −+ x + 3 a  ; +∞  5  5 1 1 a fx'( )= + >∀>⇒ 0 , xfx() 25x− 12 x − 1 5 1  a  ;+∞  f(1)= 4 ơ 5  ⇔fx( ) ≥ f (1) ⇔≥ x 1. ơ x ≥ 1 5 ơa 332−+x −≤ 26 x 2x − 1 1 3 <x ≤ 2 2
  38. 5 ơ ⇔332 −+x ≤+⇔ 2 x 6 fxgx () ≤ () (*) 2x − 1 5 1 3  f() x= 33 − 2 x + a  ;  2x − 1 2 2  −3 5 13  1 3  a fx'()= − ⇒1 fxf ( ) (1) == 8 g (1) > gx ( ) ⇒ (*) 3 aơ 1 ≤x ≤ 2 ơa (xx+ 2)(2 −− 1) 3 x +≤− 6 4 ( xx + 6)(2 −+ 1) 3 x + 2 1 x ≥ 2 ơ ⇔(x +++ 2 x 6)(2 x −−≤ 1 3) 4 (* ) • 2x−− 1 3 ≤ 0 ⇔ x ≤ 5 ⇒ (*) • x > 5 fx()(= x +++ 2 x 6)(2 x −− 13) (5; +∞ ) 11x+ 26 + x + a fx'()(= + )(213) x −−+ >∀>⇒ 0, xfx 5 () 2226x+ x + 21 x − (5; +∞ ) f (7)= 4 (*) ⇔fx () ≤ f (7) ⇔≤ x 7 1 aơ ≤x ≤ 7 2 ơa 2xxx3+ 3 2 ++< 6 1623 +− 4 x  3 2 2x+ 3 x + 6 x + 160 ≥  ⇔−2 ≤x ≤ 4. −x ≥ 4 0 ơ ⇔2xxx3 + 3 2 ++−−< 6 16 4 x 23 ⇔ fx ()23 < ()* 3 2 fx()= 2 xxx + 3 ++−− 6 16 4 x −2;4 
  39. 3(x2 + x + 1) 1 a f'( x )= + >∀∈− 0,x () 2;4 ⇒ f( x ) a 2x3+ 3 x 2 + 6 x + 16 2 4 − x (−2;4 ) f (1)= 2 3 (*) ⇔fx ( ) x1 0 , ∀ x fx( )= x4 − x + 1 » 1 a f'() x= 4 x 3 − 1 f'( x )= 0 ⇔ x = 3 4 1 f'( x ) aơ x a 3 4 1 1 1 min()f x= f () = −+> 10 34 44 3 3 4 f( x )> 0 , ∀ x ơ  2x++ 3 4 − y = 4 (1) 1.   2y++ 3 4 − x = 4 (2) 3 x+2 x = y ( 1 ) 2.  3 y+2 y = x () 2 3 3 x−3 xy = − 3 y (1) 3.  x6+ y 6 =  1 (2)  2x++ 3 4 − y = 4 (1) 1.   2y++ 3 4 − x = 4 (2)  3 − ≤x ≤ 4  2 3 − ≤y ≤ 4  2 (1) (2) a 234x+− −= xy 234 +− − y ( 3 )
  40. 3  ft()= 2 t +− 3 4 − t − ; 4  2  / 1 1 3  a f() x= + >∀∈− 0, t  ; 4  ⇒(3) ⇔fx ( ) = fy ( ) ⇔= x y 2t+ 3 24 − t 2  a x= y (1) a 2x++ 3 4 −=⇔++ xx 4 72(2 xx + 3)(4 −= )16 x = 3 9−x ≥ 0 ⇔2 − 2x2 + 5 x + 12 = 9 − x ⇔ ⇔  ơ 2  2  11 9x− 38 x + 33 = 0 x =  9  11 x = x = 3  ,  9 y = 3 11  y =  9 (1) (2) a (2x+−+ 3) (2 y 3) (4 −−− yx ) (4 ) ( 2323x+− y ++) ( 4 −− yx 4 −=) 0 ⇔ + = 0 23234x++ y + −+− yx 4  2 1  ⇔−()x y  + = 0() *  23234x++ y + −+− yx 4  2 1 + > 0 ( * ) ⇔x = y 23234x++ y + −+− yx 4 a x= y (1) a 2x++ 3 4 −=⇔++ xx 4 72(2 xx + 3)(4 −= )16 x = 3 9−x ≥ 0 ⇔2 − 2x2 + 5 x + 12 = 9 − x ⇔ ⇔   2  11 9x− 38 x + 33 = 0 x =  9  11 x = x = 3  ơ 2 ,  9 y = 3 11  y =  9 3 x+2 x = y ( 1 ) 2.  3 y+2 y = x () 2 ftttftt()=+⇒3 2 / ()3 = 2 +>∀∈ 20, t »  f( x )= y (1) ơ   f( y )= x (2) xy>⇒ fx() > fy () ⇒> yx (1) (2) xy<⇒ fx() < fy () ⇒< yx
  41. a x= y a xx3 +=⇔0 xx( 2 +=⇔= 10) xvx 0ì 2 +> 10. x = 0  y = 0 (1) (2) a xy3−+−=⇔− 3 330( xy xyxyxy )(2 +++= 2 3)0 2 y  3 y 2  ⇔−()xyx +  + +=⇔= 30  xy 2 4    x= y (1) (2) a xx3+=⇔0 xx( 2 +=⇔= 10) x 0 x = 0 ơ  y = 0 3 3 x−3 xy = − 3 y (1) 3.  x6+ y 6 =  1 (2) (1) (2) a −1 ≤x , y ≤ 1 (1)⇔fx ( ) = fy ( ) (*) ft( )= t3 − 3 t [− 1;1] a ftt'( )= 3(2 − 1) ≤ 0 ∀∈− t [ 1;1] ⇒ ft( )[− 1;1] 1 (*) ⇔x = y a (2) aa x= y = ± 6 2 ơ  1 1 x− = y − (1) 1.  x y  2 2x− xy − 1 = 0 (2)  1 1 x− = y − (1) 2.  x y  3  2y= x + 1 (2)  1 1 x− = y − (1) 1.  x y  2 2x− xy − 1 = 0 (2) x≠0, y ≠ 0 a
  42. y= x 1   (1)⇔ (x − y ) 1 +  = 0 ⇔ 1 xy  y = − .  x • y= x ơ (2)⇔x2 −= 1 0 ⇔ x =± 1 1 • y = − ơ (2) x x=1  x = − 1 ơ 2 ;  y=1  y = − 1  1 1 x− = y − (1) aa  x y  2 2x− xy − 1 = 0 (2) x≠0, y ≠ 0 1 1 fttt( )=− , ∈» \ {0} ⇒ ft/ ( ) =+>∀∈ 1 0, t » \ {0} t t2 a (1)⇔fx ( ) = fy ( ) ⇔= x y a f( t ) ơ2 a f( −1) = f ( 1) = 0  1 1 x− = y − (1) 2.  x y  3 2y= x + 1 (2) x≠0, y ≠ 0. x= y x− y 1   (1)⇔xy − + = 0 ⇔ ( xy − ) 1 +  = 0 ⇔ 1 xy xy  y = − .  x x = 1  • x= y ơ (2) ⇔ −1 ± 5 x = .  2 1 • y = − ơ (2) ⇔x4 + x +2 = 0. x −1 fxxx()=4 ++⇒ 2 fxx / ()410 = 3 +=⇔= x . 3 4 −1  3 f   =−2 > 0, lim = lim =+∞ 34  4 3 4 x→−∞ x →+∞ ⇒fx()0, > ∀∈⇒ x» xx4 ++= 20
  43. x≠0, y ≠ 0. x= y x− y 1   (1)⇔xy − + = 0 ⇔ ( xy − ) 1 +  = 0 ⇔ 1 xy xy  y = − .  x x = 1  • x= y ơ (2) ⇔ −1 ± 5 x = .  2 1 • y = − ơ (2) ⇔x4 + x +2 = 0. x • x ⇒1 x 20 xx4 ++> 20 • x≥⇒1 xxxxx4 ≥ ≥−⇒ 4 ++> 2 0 aơ (2) −+15  −− 15 x x x = 1 =  = ơ 3 ∨ 2 ∨  2 y = 1 −+15 −− 15  y=  y = 2  2 ơ  2x y =  2  1 − x  2y 1. z =  2  1 − y  2z x =  2  1 − z y3−9 x 2 + 27 x −= 27 0  2. z3−9 y 2 + 27 y − 27 = 0 x3− z 2 + z − =  9 27 27 0  2x y =  2  1 − x  2y 1. z =  2  1 − y  2z x =  2  1 − z x> y > z 2t f() t = D =» \{ ± 1 }a 1 − t 2
  44. 2(t 2 + 1) ft() = >∀∈⇒0, xDft() D (1− t 2 ) 2 xyz>>⇒ fx( ) > fy( ) > fz( ) ⇒>> yzx a ơx 0, ∀ > ft'()=⇔ 0 18 t − 27 =⇔=⇒ 0 t  2 2 3 f'() t  3 3 27 3 3  4 2 ⇒y ≥ ⇒≥ y >⇒  43 2 3 3 4 z ≥ >  3 4 2 f( x ) = y  x, y , z aơ f( y ) = z a f z= x  ( ) xy≥⇒ fx() ≥ fy () ⇒≥⇒≥ y3 z 3 yz ⇒fy() ≥ fz () ⇒≥⇒≥ z3 x 3 zx ⇒≥≥≥⇒xyzx xyz == aa xxx3−9 2 + 27 − 27 =⇒= 0 x 3 a x= y = z = 3
  45. 81 ơ 81sin10x+ cos 10 x = () * 256 ơ 3 1. (7+ 5 2)cosx −+ (17 12 2) cos x = cos 3 x ta n 2 x π π  2. e+ cx os=2 , x ∈  -;  2 2  3.2003 x+ 2005 x = 4006x + 2 x 4.3=++ 1x log(13 + 2) x ơ 3x− x 2 − 1 1  x2 x 1. log3 −+++ 322  = 2*() ( ) 5  2. x−3log x −+ 5 log xx −=+ 3  2 ( ) 3( ) 5 ( )  3 3  x+ x − 4 3. log2 x +  + 2 = 2 2  ơ  2y− 1 x+ x −+=2 x 23 + 1 1.  (x , y ∈ » ) y+ y2 −+= y x − 1 +  2 23 1  2xy−−+ 12 xy 21 xy −+ (1+ 4 )5 = 1 + 2 (1) 2.  yx3+++ yx 2 + =  4 1 ln( 2 ) 0 (2)  x y e =2009 −  y 2 − 1 ơ  1 2 a x () ey =2009 −  x 2 − 1 x>1, y > 1 ln(1+−x ) ln(1 + yxy ) =− ( 1 ) ơ  x2− xy + y 2 = 25 02() ơ xx3+3 −+ 3ln( xx 2 −+= 1) y   1. yy3+3 −+ 3 ln( yy 2 −+= 1) z  zz3+3 −+ 3 ln( zz 2 −+= 1) x 
  46.  xx2 −+2 6log(6 −= yx )  3  2. yy2 −+ −= zy  2 6log(63 )   zz2 −+2 6log(6 −= xz )  3 t=sin2 x ; 0 ≤ t ≤ 1 81 ơ ()*⇔ 81t5 +−= (1 t ) 5 , t ∈  0;1  256   5 5 ft( )= 81 t + (1 − t ) 0;1  a 4 4   ft'()= 5[81 t −− (1 t )],t ∈  0;1   4 4 81t= (1 − t ) 1 f'() t=⇔ 0  ⇔= t t ∈   4  0;1  1 81 a f() t≥ f () = 4 256 1 1 1 π ơ t=⇔sin2 x =⇔ cos2 x =⇔=+ xkkZπ ( ∈ ) 4 4 26 3 1. (7+ 5 2)cosx −+ (17 12 2) cos x = cos 3 x 3 ⇔+(1 2)3 cosx −+ (1 2) 4 cos x = 4cos 3 x − 3cos x 3 ⇔+(1 2)3 cosx + 3cosx = 4cos 3 x ++ (1 2) 4 cos x t f() t=(1 + 2 ) + t » a t ft'() =+( 1 2ln1) ( + 2) +>∀∈⇒ 10, tft» () » a f(3 cos xf) = ( 4 cos 3 x ) πk π ⇔()3 cosx = 4 cos3 xxx ⇔ cos 3 =⇔=+ 0 , k ∈ » 6 3 ta n 2 x π π  2. e+ cx os=2 , x ∈  -;  2 2  ta n 2 x π π  fxe()= + cx os x ∈ - ;  2 2  a 2 tan x 3  12 2e− c os x fx'()2tn.= ax eta n x − sin x = sin x   2 3  cosx c os x   
  47. 2 2eta n x ≥ 2 > c os3 x > 0 a f'( x ) a sin x a f( x )≥ f (0) = 2 ơ x = 0 3.2003 x+ 2005 x = 4006x + 2 f( x )= 2003x + 2005 x − 4006 x − 2 » a f'( x )= 2003x ln 2003 + 2005 x ln 2005 − 4006 f''( x )= 2003x ln2 2003 + 2005 x ln 2 2005 >∀∈ 0 x » ⇒f"( x ) = 0 ⇒f'( x ) = 0 ơ f( x ) = 0 a f(0) = f ( 1) = 0 ơa x=0, x = 1 x 4.3=++ 1x log(13 + 2) x 1 x > − 2 x x x ơ ⇔+=++3xxx 1 2 log(13 +⇔+ 2) 3 log3 3 =++ 1 2 xx log(1 3 + 2)*( ) 1 ft()= t + log t (0; +∞ ) a ft'() =+ 1 > 0, t >⇒ 0 ft() 3 t ln 3 (0; +∞ ) ơ (*) ⇔ffx (3)x = (12) + ⇔=+⇔−−= 3 x 2 x 1 3 x 2 x 10 ( ) fx()=−−⇒ 3x 2 x 1 fx '() = 3ln32 x −⇒ fx "() = 3ln3 x 2 > 0 ⇒f( x ) = 0 a f(0)= f ( 1) = 0 ơa x=0, x = 1 3x− x 2 − 1 1  x2 x 1. log3 −+++ 322  = 2*() ( ) 5  xx2 −320 +≥⇔≤∨≥ xx 1 2 u= x2 −+3 x 2, u ≥ 0 1−u2 1  1 u2 ơ ()()*⇔ log3 u ++ 2 =⇔ 2 log3 () u ++ 2  .5 =≥ 2, u 0 () 5  5 1  2 f u=log u + 2 + .5 u a 0; +∞ ()()3    ) 5  1 1 2 a fu'()= + 5.ln5.2u uufu >∀≥⇒ 0, 0 () a 0; +∞ ) (u + 2)ln3 5  f(1) = 2 ⇒ u = 1 ơ (* * )
  48.  3− 5 x = xx2−321 + = ⇔ xx 2 − 310 + = ⇔  2  3+ 5 x =  2 2. x−3log x −+ 5 log xx −=+ 3  2 ( ) 3( ) 5 ( )  x > 5 ơ x−3log x −+ 5 log xx −=+ 3  2 ( ) 3( ) 5 ( )  x + 2 x x ⇔log3()() −+ 5 log 5 −= 3 x − 3 fx x x ( ) =log3( −+ 5) log 5 ( − 3 ) (5; +∞ ) 1 1 fx'() = + >∀>⇒ 0, xfx 5 () (5; +∞ ) ()x−5ln3() x − 3ln3 x + 2 g() x = (5; +∞ ) x − 3 −5 gx'= ⇒ 0, x 5 gx 5; +∞ () 2 () ( ) ()x − 3 g(8) = f ( 8) = 2 ơ x = 8 3   x+ x − 3 4 3. log2 x +  + 2 = 2 2  x ≥ 0 t= x, t ≥ 0 3   t2 + t − 3 4 ơ ⇔log2 t ++  2 −=≥ 20, t 0 2  3 3  t2 + t − f t=log t ++ 24 − 2 a 0; +∞ () 2    ) 2  3 1 t2 + t − a ft'() = ++ (2 t 1)24 .ln 2 >∀≥⇒ 0, tft 0 () a 3  t +  .ln 2 2  1  1 0; +∞ ) f  =0 ⇒ t = aơ f( t ) = 0 2  2 1 1 1 t=⇒ x =⇒= x 2 2 4
  49.  2y− 1 x+ x −+=2 x 23 + 1 1.  (x , y ∈ » ) y+ y2 −+= y x − 1 +  2 23 1 u= x −1, vy =− 1  2 v u+ u +1 = 3 (I )  (II ) +v v 2 += u  1 3 fx() = x + x 2 + 1 g( x ) = 3x ∀x ∈ » a x x2 +1 + x x+ x f'() x =+ 1 = > ≥∀∈0, x » ⇒ f( x ) ∀x ∈ » x2+1 x 2 + 1 x 2 + 1 g( x ) = 3x ∀x ∈ »  2 v uu++=1 3  fugv()() = ⇔  ⇒+=+fu()()()() fv gu gv v+ v 2 + = u  fv()()= gu  1 3  uu>⇒ fu( ) > fv( ) ⇒ gv( ) > gu( ) ⇒> vu ơ v> u 2u  u 2 uu++=13  13( = uu +− 1 )(1) ()II ⇔ ⇔  uv= uv =   gu() =3(u u2 + 1 − u ) ∀u ∈ » u  a gu'()3ln3(=u u2 +−+ 1 u )3 u  − 1    u2 + 1    1  guuu'()3=u 2 +− 1 ln3 −  >∀∈ 0, u »       u2 + 1  g( u ) ∀u ∈ » g(0) = 1 ⇒ u = 0 a (1) (II) ⇔u = v = 0 (I )⇔ x = y = 1  2xy−−+ 12 xy 21 xy −+ (1+ 4 )5 = 1 + 2 (1) 2.  yx3+++ yx 2 + =  4 1 ln( 2 ) 0 (2) t t  1  4 t t=2 x − y ơ(1) 5+   = 12.2 + ()* 5  5   
  50. t t  1  4 t f() t =5  +   g( t ) =1 + 2.2 5  5    t t  1  4 t f() t =5  +   g( t ) =1 + 2.2 5  5    f(1) = g( 15) =⇒= t 1 a (*) a t = 1 aơ ft( ) = gt( ) •>⇒ t1 ftgt( ) t 1 ơ (*) • gt( ) ⇒ 0 yy2++1 yy 2 ++ 1 ⇒ f( y ) f(− 1) = 0 (* * ) y = − 1 x = 0 a  y = −  1 t ft()()= egtt , = (1, +∞ ) a t 2 − 1 fte'( ) =t > 0, ∀> t 1 ⇒ ft( ) (1, +∞ ) −1 gt/( )= ⇒ 0, t 1 gt() (1, +∞ ) (t 2− 1) 3  x y e =2009 − 2  y − 1  fx()()+ gy = 2009 ơ  ()1 ⇔  x fy+ gx = 2009 ey =2009 −   ()()  x 2 − 1 ⇒fx( ) + gy( ) = fy( ) + gx( ) xy>⇒ fx( ) > fy( ) ⇒ gy( ) yx ơ y> x
  51.  x y e =2009 −  x x 2  e + −2009 = 0 y − 1 1⇔ 2 2 x ()  x − 1 () y x= y e =2009 −   x 2 − 1 x h() x= e x + − 2009 (1; +∞ ) a x 2 − 1 1 3 1 hxe'() =−x ,'' hxe() =+x .20 x > 3 2 5 ()x2 −1() x 2 − 1 limhx( ) = +∞ ,lim hx( ) = +∞ x →1+ x →+∞ h( x ) (1; +∞ ) ơ a x > h x (2) 2 1 0 1 ( 0 ) 0 =2:2() =+− 20090 1, y > 1 x>−1, y >− 1 ơ (1) ⇔ ln(1 +−=xx ) ln(1 +− yy ) ( 3 ) ft( ) =ln(1 + tt ) − (−1; +∞ ) −t a f'( t )= , ∀ t ∈() − 1; +∞ f'() t= 0 ⇔ t = 0 1 + t • ft'( ) >∀∈− 0, t( 1;0 ) ⇒ ft( ) (−1;0 ) • ft'( ) < 0, ∀∈ t( 0; +∞⇒) ft( ) (0; +∞ ) ơ (3) ⇔fx( ) = fy( ) ⇔= xy • x= y ơ (225.) ⇔xxxx2 − +=⇔=⇒= 2 0 x 0 y 0 ơ (x; y ) = ( 0;0 ) xx3+3 −+ 3ln( xx 2 −+= 1) y   1. yy3+3 −+ 3 ln( yy 2 −+= 1) z  zz3+3 −+ 3 ln( zz 2 −+= 1) x  f( x ) = y  ơ f( y ) = z  f z= x  ( ) a (x; y ; z ) a ftt( )=+−+3 3 t 3 ln( tt 2 −+ 1), t ∈ »
  52. 2t − 1 a ftt'()3=++2 3 >∀∈⇒ 0, t» ft() ∀t ∈ » 2t2 − t + 1 x= max{ xyz ; ; } y= fx() ≥ fy () =⇒= z z fy () ≥ fz () = x x= y = z ơ xx3+2 −+ 3ln( xx 2 −+= 1) 0 gxx( ) =+−+32 x 3 ln( xx 2 −+ 1), x ∈ » g( x ) » g (1) = 0 ơg( x ) = 0 x = 1 x= y = z = 1  xx2 −+2 6log(6 −= yx )  3  yy2 −+ −= zy 2. 2 6log(63 )   zz2 −+2 6log(6 −= xz )  3  x  −y = log3 (6 )  2  x−2 x + 6 fy()= gx ()  y  ⇔ −=z ⇔= fzgy log(6)3  ()() 2 y−2 y + 6  fx= gz  () ()  z  −x = log3 (6 )  2  z−2 z + 6 t ft= − tgt = t ∈ −∞ ( ) log3 (6 ) ; ( ) , ( ;6) t2 −2 t + 6 1 a ft'( )=− 0, ∀∈−∞⇒ t ( ;6) gt() (−∞ ;6) 3 (t2 −2 t + 6 ) a (x; y ; z ) a x= y = z aa x −=x ⇔= x log3 (6 ) 3 x2 −2 x + 6 a x= y = z = 3 A
  53. fx()= gx ()  1 2 fx()= gx () a  2 3  fx()= gx ()  n 1 f g A x x x a A , (1 , 2 , ,n ) x= x = = x 1 2 n f g ơ A x x x a A , (1 , 2 , ,n ) x= x = = x x= x = = x n  1 3n − 1 n 1 2 n x= x = = x  2 4 n ơ sinx− sin y = 3 xy − 3 (1)   π 1. x+ y = (2)  5 x, y > 0 (3)  log (1+ 3 cosx ) = log (sin y ) + 2  2 3 2.  log (1+ 3 siny ) = log (cos x ) + 2  2 3 sinx− sin y = 3 xy − 3 (1)   π 1. x+ y = (2)  5 x, y > 0 (3)  π  ()()2 , 3⇒x , y ∈  0;  5  (1) ⇔ sinxx −= 3 sin yy − 3 ( * ) π  π  ft() =sin t − 3 tt , ∈  0;  a ft'() = cos t −<∈ 3 0, t 0;  ⇒ ft() 5  5  π t ∈ (0; ) (*) ⇔fx( ) = fy( ) ⇔= xy 5 π x= y a (2) a x= y = 10 π π  ()x; y =  ;  a 10 10  log (1+ 3 cosx ) = log (sin y ) + 2  2 3 2.  log (1+ 3 siny ) = log (cos x ) + 2  2 3
  54. cosx > 0  y > sin 0 log(1+ 3u ) = log() v + 21  2 3 ( ) u=cos xv ; = sin y a  log(1+ 3)v = log() u + 22  2 3 () a log(1++ 3)u log u = log(1 ++ 3) v log vfufv ⇔= () () * 3 33 3 ( ) ft= + t + t f t u v ( ) log3 (1 3 ) log 3 ( ) (*) ⇔ = 1+ 3u 1 a (1) a log(13)log+−u u =⇔ 2 =⇔= 9 u 3 3 u 6 1  y=α + k 2 π sin y =    1 6 y=π − α + k π ⇔ ⇔   2 sinα= cos β = 1  6 cos x = x= ±β + m π 6  2  2 2 x 2 + 1 ey− x = ơ  y 2 + 1 3log(xy++= 2 6) 2log( xy +++ 2) 1  3 2  2 2 x 2 + 1 ey− x =  y 2 + 1 3log(xy++= 2 6) 2log( xy +++ 2) 1  3 2 x+2 y + 6 > 0  x+ y +2 > 0 2 2 2 2 y− x x + 1 22x + 1 2 2 ln 2 aơ e = a yx−=ln = ln x +− 1ln y + 1 2 2 ( ) ( ) y + 1 y + 1 ⇔++x21ln( xy 22 +=++ 1) 1ln( y 2 + 1*) () ơ (*) fx( 2+1) = fy( 2 + 1 )() ft( ) =ln tt + a 1; +∞ ) a 1 ft'= + 1 > 0, ∀≥ t 1 ⇒ ft a 1; +∞ () t ()  ) ( ) ⇔x2 += 1 y 2 +⇔ 1 xy =± • x= − y aơ xy++= xy +++ a 3log(3 2 6) 2log( 2 2) 1
  55. x − 1 2u x +2 = 3 3log(x+= 2) 2log( x +=⇒ 1) 6 u  3 2 x + = 3u  1 2 u u 3u 2 u 1  8 ⇒+=1 2 3 ⇔ +  = 1 9  9 u u 1  8 g() u = +  » g (1) = 1 u = 1 9  9 u u 1  8 aơ +  = 1 9  9 u=1 ⇒( x ; y ) = ( 7;7 ) ơ x2+ y 3 = 29 (1) aơ x; y a  ( ) logx .log y = 1(2)  3 2 2 3 x+ y = 29( 1 )  a logx .log y = 1 2  3 2 () (x; y ) a x>1, y > 1 ( 3 ) 1 x t t x = t y t log3 = , > 0 (3) 3 ơ (2) = 2 1 ơ ()1⇔ 9t + 8t = 294 () aơaơ (4) 1 1 −29 8t .ln 8 f() t =9t + 8 t (0; +∞ ) a f'() t = 9.ln9t − . t 2 1 1 (0; +∞ ) y = 8t .ln 8 y = ơ t 2 1 8t .ln 8 (0; +∞ ) y = a f' ( t ) t 2 (0; +∞ ) 1  f f =−256 −< t a f t '  . '(1) 18(ln9 ln2 )(ln27 ln16) 0 ∃0 ∈ (0; 1 ) '( 0 ) = 0 2 
  56. aa f( t ) (0; +∞ ) t 0 t0 1 +∞ f' ( t ) − 0 + f( t ) ∞ +∞ −12 f t ( 0 ) aơ (4) aơơ a ư ưaa f( x; m ) = 0 x∈ I (*) • (*) fx( ) = fm( ) • y= f( x ) I • ơa a m ơ x+3 x2 + 1 = m fx( ) = x +3 x 2 + 1 y= m fx( ) = x +3 x 2 + 1 » 3x 313 x2 + + x a f'() x = 1 + = 31x2+ 31 x 2 + x < 0 fx'= 0 ⇔ 313 x2 + = − x ⇔  () x2+ = x 2 3 1 9 x < 0  −6 − 6  6 ⇔ ⇔=x, f   =  ±1 ± 6   x = = 6 6  3  6 6 6 6 a f() x ≥ f( x) = m m ≥ ơ 3 3 a m ơ 4 x2 +1 − x = m ( 1 )
  57. 4 2 fx( ) = x +1 − x a 0; +∞ )   1x 1  a f'() x = −  0 ⇒ fx a 34 6 3 ( ) ( ) 2 x2 x 4 ()x+1x 4 () x + 1 0; +∞ limf ( x )= 0 0<f ( x ) ≤∀∈ 1, x  0; +∞  ) x → + ∞  ) ơ (1) a 0;+∞) ⇔ 0 <m ≤ 1 a m ơ (4m− 3) x ++ 33( m − 41) −+−= xm 10,2( ) −3 ≤x ≤ 1 3x++ 341 −+ x 1 ơ (2) ⇔m = 4x++ 331 −+ x 1 2 2 2 2 x+3  1 − x  x++−=⇔314 x   +  = 1 ( ) ( )    2  2  π  ϕ ϕ ∈0;  ,t = t a n ; t ∈  0;1  a 2  2 2t 1 − t 2 x +3 = 2 sinϕ = 2 1−x = 2 cosϕ = 2 1 + t 2 1 + t 2 3x++ 341 −+ x 1 −++ 7129 tt2 m= ⇔= m = ft()(), 3 4x++ 331 −+ x 1 −5t2 + 16 t + 7 −7t2 + 12 t + 9 f( t ) = t ∈ 0;1  a −5t2 + 16 t + 7   −52t2 − 8 t − 60 9 7 ft'( )= <∀∈⇒ 0, t 0;1  ft f(0)= ; f (1) = 2   () [ ] ()−5t2 + 16 t + 7 7 9
  58. aơ (2) ơ (3) t ∈ 0;1  7 9 ≤m ≤ 9 7 a m ơ 2 2 x−2 x + 24 ≤ x − 2 xm + −4;6  2 t= x −2 x + 24 ∀∈−x4;6  ⇒∈ t  0;5 2 a m ơ t+ t −24 ≤ m t ∈ 0;5  2 ft( ) = t + t − 24 0;5    a ftt'()= 2 +>∀∈ 1 0, t 0;5  ⇒ ft( ) 0;5  ơ 0;5  maxftmf ( )≤⇔ (5) ≤⇔≤⇔≥ m 6 mm 6   t∈0;5  ư a ABC 1 13 cosA+++ cos B cos C = a ABC cosA+ cos B + cos C 6 A B C 3 tABC=++=+cos cos cos 1 4 sin sin sin ⇒ ∀∈ 0, t 0; ⇒ ft a 0; () 2   ()   t 2  2  3 t 0 2 f' ( t ) + 13 f( t ) 6 2
  59. 13 aa 2 f ( 0 ) x > 0 x = 0 0; +∞ ) aa 0 • a D f( x 0 ) f • D
  60. • x a f x f x a f 0 ( 0;( 0 ) ) f x f x f x 0 0 '( 0 ) = 0 • f ' 0 x 0 f x 0 • • a 0 • x x f x 0 ( 0;( 0 ) ) y= x y= x 3 f a b a x a x ( ; ) 0 ( ; 0 ) x b ( 0; )  fx'( 0) 0, x ∈ xb ;  ()()0 0 aơ x a x0 x 0 x a x0 b f' ( x ) − + f( x ) f( a ) f( b ) f( x 0 )  fx'( 0) > 0, x ∈ ( ax ; 0 ) b x )  0 f' ( x ) fx'< 0, x ∈ xb ;  ()()0 0 ơa x a x0 x 0 x a x0 b f' ( x ) + − f x f x ( ) ( 0 ) f( a ) f( b ) f a b a x f x f ( ; ) 0 '( 0 ) = 0 a 0 x0
  61. a f x f x ) ''( 0 ) 0 0 f a x= x 0 a iii x0 1−x khi x ≤ 0 f( x ) =  x = 0 x khi x >  0 x = 0 O a uy • f' ( x ) • xi ( i = 1,2,3 ) 0 x x x • a f' ( x ) f' ( x ) a 0 0 uy • f' ( x ) • xi ( i = 1,2,3 ) aơ f'( x ) = 0 • xi f''( x i ) . − f''( x i ) 0 xi a 1 5 1. yfx=() = xx3 −−+ 2 3 x 3 3 2. yfx=( ) =+ x3 3 x 2 ++ 3 x 5 1 5 1. fx() = xx3 −− 2 3 x + 3 3 » a fx'( ) = x2 − 2 x − 3 fx'( ) = 0 ⇔ x =− 1, x = 3
  62. x −∞ −1 3 +∞ f' ( x ) + 0 − 0 + 10 f( x ) +∞ 3 22 −∞ − 3 10 22 x=−1, f () − 1 = x=3, f () 3 = − 3 3 f''( x) = 2 x − 2 10 f ''(− 1) =− 4 0 x=3, f () 3 = − 3 2. yfx=( ) =+ x3 3 x 2 ++ 3 x 5 » a yxx'= 32 + 6 += 3 3( x + 1) 2 ≥∀⇒ 0 x y ' a y '= 0 a a 1. yfx=( ) =−+ x4 6 x 2 − 8 x + 1 2. yfx=( ) =−+ x4 2 x 2 + 1 1. yfx=( ) =−+ x4 6 x 2 − 8 x + 1 » a yxx'=− 43 + 12 −=− 8 4( xx − 1) 2 ( + 2) x = 1 y'= 0 ⇔ − 4( x − 1)2 ( x + 2) = 0 ⇔  x = −  2 x −∞ −2 1 +∞ y ' + 0 + 0 − 25 y −∞ −∞ x = − 2 a y(− 2) = 25 2. yfx=( ) =−+ x4 2 x 2 + 1 » a y'=− 4 x3 + 4 x =− 4( xx 2 − 1)
  63. x = 0 y'0= ⇔ − 4( x x 2 − 1)0 = ⇔  x = ±  1 x −∞ −1 0 1 +∞ y ' + 0 − 0 + 0 − 2 2 y −∞ 1 −∞ x = ± 1a y(± 1) = 2 x = 0 a y(0)= 1 a x = 0 a y ' aaa ơ y '= 0 aơ aơ y '= 0 a a 1. y= fx( ) = x 2. y= fx( ) = xx( + 2 ) 3. y= fx( ) = xx( − 3 ) 1. y= fx( ) = x » x khi x ≥ 0 y =  −x khi x 0 a y =' =  − khi x <  1 0 x −∞ 0 +∞ y ' y +∞ +∞ 0 x=0, f ( 0) = 0 x( x+2) khi x ≥ 0 2. y= fx() = xx() += 2  −xx + khix <  ()2 0 »
  64. 220x+ > khi x > 0 a y ' =  −x − khi x 0  a y ' =  2 x 3 − x  +−x >0 khi x < 0 2 −x y'= 0 ⇔ x = 1 x −∞ 0 1 +∞ y ' + − 0 + y 0 +∞ −∞ −2 x=0, f ( 0) = 0 x=1, f ( 1) = − 2 a 1. yfx=( ) = x 4 − x 2 2. yfx=() =− 2 x x 2 − 3 3. yfx=() =−+ x3 3 x 2
  65. 1. yfx=( ) = x 4 − x 2 −2;2  4− 2 x 2 a y'= , x ∈() − 2;2 4 − x 2 y'=⇔ 0 x =− 2, x = 2 y 'aơ x a − 2 x = − 2, f (−2) = − 2 y 'ơa x a 2 x = 2, f ( 2) = 2 x −2 − 2 2 2 y ' − 0 + 0 − y 0 2 −2 0 2. yfx=() =− 2 x x 2 − 3 a(−∞ ; − 3] ∪ [3; +∞ ) x2 x2 − 3 − x ay'2=− = , x ∈−∞−∪( ;33;) ( +∞ ) x2−3 x 2 − 3   x ∈−∞−( ; 3) ∪( 3; +∞ )  0≤x < 3 y '=⇔ 0 ⇔ ⇔=x 2   2 2 x2 − = x  4(x− 3) = x 2 3  x = ± 3 x −∞ − 3 3 2 +∞ y ' + − 0 + y +∞ −∞ 3 x=2, y (2) = 3 3. yfx=() =−+ x3 3 x 2 a(−∞ ;3] −3(x2 − 2 x ) a y'= , x < 3, x ≠ 0 2−x3 + 3 x 2 y'= 0 ⇔ x = 2 x=0; x = 3
  66. x −∞ 0 2 3 y ' − + − y +∞ 2 0 0 x=2, y (2) = 2 x=0, y (0) = 0 x = ± 3 (a ; b ) aaaa ơ x = 3 a x = 0 a aa 1. yfx=( ) = 2sin2 x − 3 2. yfx=( ) =− 3 2cos x − cos2 x 1. yfx=( ) = 2sin2 x − 3 » a y'= 4 cos2 x π π y'=⇔ 0 cos2 x =⇔=+ 0 xkk , ∈ » 4 2 y''= − 8 sin 2 x , π π   π  −8 khi k = 2 n y''+ k =− 8 sin + k π =      khi k= n + 4 2   2  8 2 1 π π  x=+ nyπ; + n π  =− 1 4 4  π ππ π  x=++()21; n y ++() 21 n  =− 5 4 24 2  2. yfx=( ) =− 3 2cos x − cos2 x » a y'= 2sin x + 2sin2 xx = 2sin( 1 + 2cos x ) sinx= 0  x = k π   y '=⇔ 0 1 2π ⇔ 2 π , k ∈ » cosx=−= cos  x =±+ k 2 π 2 3  3 y''= 2 cos x + 4 cos2 x
  67. 2π  2 π 2π 2π  1 y''±+ k 2π  = 6 cos =− ∀∈ 4 0, k » xkyk=π , ( π) = 2( 1 − cos k π )  3 1+x sin2 x − 1  , x ≠ 0 f( x ) =  x a  x = 0 , 0 x = 0 x = 0 fxf( )− (0)3 1 + xx sin2 − 1 f '0() = lim = lim x→0x x → 0 x 2 xsin 2 x f '() 0= lim x →0 2 2 23 2  x3 ()1+ xx sin ++ 1 xx sin + 1    sinx 1 f'0limsin.() = x .= 0 x →0 x 2 3 ()1+xx sin2 ++3 1 xx sin 2 + 1 sin 2 x x ≠ 0 a f x = ⇒≥=f x0 f 0 . () 2 ()() 3 ()1+xx sin2 ++3 1 xx sin 2 + 1 f( x ) » f( x ) x = 0 I a 1. y= − x3 + 3 x 2 2. y= x4 − 4 x 3 + 1 1. y= − x3 + 3 x 2 a y'3=− xxy2 + 6 ⇒ '0 =⇔= xx 0;2 = yxy"=−+⇒ 6 6 "(0) => 6 0 ; y "(2) =−< 6 0 x = 2 ay(2)= 4 x = 0 a y(0)= 0 2. y= x4 − 4 x 3 + 1 x = 0 a yx'4=3 − 8 x 2 = 4( xx 2 − 2) ⇒ y '0 = ⇔  x =  2
  68. x −∞ 0 2 +∞ y ' − 0 − 0 + +∞ +∞ y −15 x = 2 a y(2)= − 15 f D ⇔ ∃x0 ∈ D aaa a x0 x0 f'( x ) a x0 f"( x 0 )≠ 0 f'( x ) aaaa ⇔ ơ f'( x ) a m y= mx3 +3 x 2 + 12 x + 2 x = 2 » a y'3= mx2 ++⇒= 612 x y "6 mx + 6 y '(2)= 0 x =2 ⇔  y <  "(2) 0 12m + 24 = 0 ⇔ ⇔=−m 2 m + < 12 6 0 aa x = 2 y'(2)= 0 ⇔ m =− 2 m = − 2a y'=− 3( 2 x2 + 2 x + 4) a x = 2 x2 + mx + 1 1 . a m y= f() x = x+ m x = 2. 2 . a m yfxx=( ) =++3( m3) x 2 +− 1 m x = − 1. 1. D=» \ { − m } x2+2 mx + m 2 − 1 a y'= , x ≠ − m 2 ()x+ m
  69. m = − 3 x = 2 y'20= ⇔ m2 + 4 m + 30 = ⇔  () m = −  1 x2 −6 x + 8 m = − 3a y'= , x ≠ 3 2 ()x − 3 x = 2 y '= 0 ⇔  x =  4 x −∞ 2 3 4 +∞ y ' + 0 − − 0 + y 1 +∞ +∞ −∞ −∞ 5 aa x = 2 m = − 3 ơ m = − 1 D=» \ { − m } x2+2 mx + m 2 − 1 a fx'= , xm ≠ − () 2 ()x+ m 2 y''= , x ≠ − m 3 ()x+ m  1 1− = 0 2  2 m+4 m + 3 = 0 y '() 2= 0  ()2 + m  x = 2 ⇔  ⇔≠−  m 2 y ''2< 0 2 () < 0  3 m < − 2  + m  ()2 m=−1 ∨ m =− 3 ⇔ ⇔=−m 3 m < −  2 m = − 3 2. » a y'32=+ x2 ( m + 3) xxxm =( 326 ++ ) x = 0  y '= 0 ⇔ 2m + 6 x = −  3
  70. 2m + 6 x −∞ − 0 +∞ 3 y ' + 0 − 0 + y 2m + 6 3 x=−⇔−1 =−⇔ 1 m =− . 3 2 x2 + mx − 2 m ∈ » y = mx − 1 1  » \   m  m = 0 y= x 2 −2 ⇒ 1 m ≠ 0 ∀x ≠ m mx2 −2 x + m a y ' = ơ mx2 −2 x + m = 0 a (mx − 1) 2 1−m2 > 0 1  ⇔ 1 ⇔− ∀ m ( ) ' 'g ( 110,) ∀m x m x m g( x ) = 0 2 1= −1, 2 =+ 1 x −∞ m − 1 m m + 1 +∞ y ' + 0 − − 0 + y +∞ +∞ −∞ −∞
  71. x y 'ơa a x1 = m − 1 x1 = m − 1 x y 'aơ a x2 = m + 1 x2 = m + 1 y=+ x44 mx 3 + 3( m + 1) x 2 + 1 m ∈ » 1. a 2. » a y'=+ 4 x3 12 mx 2 ++= 6( m 1) x 2 xx (2 2 +++ 6 mx 3( m 1)) x = 0 y '= 0 ⇔  fx= x2 + mx + m +=  ()2 6 3 30 y a x1, x 2 ≠ 0 y ' aa 0,x1 , x 2 a y x = 0 y ' − a + a y y ' aa x = 0 x = 0 a 1. a y a 2  17− 17 + ∆=' 3(3m − 2 m −> 2) 0  m ⇔ ⇔  3 3 y(0)≠ 0  m ≠ −   1 2. a 17− 17 + ⇔ a ⇔ ≤m ≤ 3 3 ơ y= ax4 + bx 2 + ca ( ≠ 0) x = 0 a y'4= ax3 + 2 bx = xax (4 2 + b ) ⇒ y '0 = ⇔  ax2 + b = 4 0 (1) b ≠ 0 a ⇔a ⇔  ab 0 a 0 x =0 ⇔ ⇔  a > 0 a < 0 y= b = (0) 0  0 y= ax4 + bx 3 + cx 2 + d
  72. x = 0 a y'4= ax3 + 3 bx 2 + 2 cx ⇒ y '0 = ⇔  ax2 + bx + c = 4 3 2 0 (2) aa 9b2 − 32 ac > 0 ⇔  a a > 0 c ≠  0 a 0 a ⇔4 0 m 2b> 0 b > 0  ''0() 0   f'10( ) =  32 abc ++= 0 f x x = 1 ⇔  ()2 ( ) f < 6a+ 2 b < 0  ''1() 0 