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2016-2017学年高中数学人教A版选修2-2课时训练:1-3 导数在研究函数中的应用1-3-1 含答案 精品

1.3 1.3.1

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1

2

3()

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x1x2 f(x1) f(x2) yf(x) f(x1) f(x2)









[]



(1)(ab)





f(x)>0



f(x)<0



f(x)0



(2)(ab)



f(x)0 f(x)0 f(x)0



1 f(x)sinx x2



f(x)xcos

xsin x2

x

x2

cos x<0sin x0xcos xsin x<0

f(x)<0f(x)2

(1)



(2)f(x)>(<)0 f(x)()f(x)

() f(x)()0.

1 f(x)lnxx(0e)



f(x)lnx xf(x)x1xx2ln

x1xl2n

x .

0<x<eln x<ln e1. f(x)1xl2n x>0

f(x)(0e)



2 (1)f(x)2x33x236 x1

(2)f(x)sin xx(0<x<) (3)f(x)3x22ln x (4)f(x)x33tx. (1)f(x) 6x26x36 f(x)>0 6x26x36>0 x< 3 x>2 f(x)0 3<x<2. f(x)(3)(2) (3,2) (2)f(x)cos x1. 0x cos x10 f(x)(0) (3)(0) f(x)6x2x23x2x1. f(x)>0 23x2x10



33x0



x

3 3.

x0x

3 3.

f(x)0 23x2x10



x

33

0x

3 3.

x00x

3 3.

f(x) 33

0 33. (4) f(x)3x23t f(x) 0 3x23t0 x2t. t0 f(x) 0 ()

t>0 x2t x t x t f(x)0 tx t. f(x)( t)( t) ( t t) (1) f(x)(2) f(x)(3) f(x)0 f(x)0(4) f(x)0 f(x)0 2 (1)f(x)x2ln x (2)f(x)x3x2x. (1) f(x)(0) f(x)2x1x f(x)2x1x0 x0 x 22 f(x) 22 f(x)0 x 22 x(0) f(x)0 22. (2)f(x)3x22x1(3x1)(x1) f(x)0 x13 x1 f(x)0 13x1 f(x)13(1)131. 3 f(x)x2ax(x0 aR) f(x) x[2) a f(x)2xxa22x3x2 a.

f(x)[2) f(x)0 x[2) 2x3x2 a0 x[2)x2>0 2x3a0a2x3 x[2) a(2x3)min.x[2)y2x3 (2x3)min16a16. a16 f(x)2x3x2 160(x[2)) f(2)0a (16] f(x) I () f(x)0( f(x)0) I 3 f(x)ax3x a f(x)3ax21 f(x) f(x)3ax210 02413a0a0. a (0)
1 f(x)xln x (0,6)( ) A B C01e1e6 D01e1e6 A x(0,6)f(x)11x0 f(x)(0,6) 2f(x) yf(x) yf(x) yf(x) ( )

D

x0 f(x)0 f(x) 0

x2 f(x)0 f(x) x2 f(x)0 f(x)

D

3 f(x)x3ax2x6 (0,1) a ( )

A[1)

Ba1

C(1]

D(0,1)

A

f(x)3x22ax1 f(x)(0,1)

3x22ax10 (0,1)f(0)0 f(1)0a1.

4 yx24xa ________________

(2) (2)

y2x4 y0 x2 y0 x2

yx24xa (2)(2)

1 2 f(x) (1) f(x) (2) f(x) (3) f(x) f(x)0 f(x)0

(4)(3) f(x).



1 x(ab) f(x)>0f(x)(ab)

( )

A

B

C

D



A f(x)x3 (1,1) f(x)3x20(1<x<1)

A.

2 y12x2ln x (

)

A(0,1)

B (0,1) (

1)

C(1)

D()

A

y12x2ln x (0)yx1x y<0 x1x<0

0<x<1 x<1.

x>00<x<1 A.

3 f(x)x3ax2bxc abc a23b0 f(x)( )

A

B

C

D

A

f(x)3x22axb f(x)0 4(a23b)0 f(x)0 f(x)

4(0)( )

Aysin x

Byxe2

Cyx3x

Dyln xx

B

ysin x (0) A yxe2

e2 yxe2 (0)

Cy3x213x 33x 33

33 33

33 33 Dy1x1 (x0) (1) (0,1) B. 5 yf(x)323 yf(x) yf(x) f(x)0 ________

131[2,3) 6 yln(x2x2)________ (1) f(x)x22xx1 2 f(x)0 x1 12x2 (1)(2)(1) 7 f(x)x3ax8 (5,5) yf(x) f(x)3x2a. (5,5) yf(x)5,5 3x2a0 a75. f(x)3x275

f(x)0 3x2750 x5 x5 yf(x) (5)(5) 8 f(x) yf(x)( )
A f(x) f(x) A. 9 f(x)g(x)[ab] f(x)g(x) axb ( ) Af(x)g(x) Bf(x)g(x) Cf(x)g(a)g(x)f(a) Df(x)g(b)g(x)f(b) C f(x)g(x)0 (f(x)g(x))0 f(x)g(x) [ab] axb f(x)g(x)f(a)g(a) f(x)g(a)g(x)f(a) 10(2013) f(x)x2ax1x12 a ________ [3)

f(x)x2ax1x12 f(x)2xax120 12 ax122x 12 h(x)x122x h(x)x232 x12h(x)0 h(x) h(x)h123 a3. 11 (1)yxln x (2)yln(2x3)x2. (1)(0)y11x y0 x1 y0 0x1. yxln x (1)(0,1) (2) yln(2x3)x2 32. yln(2x3)x2 y2x2 32x4x22x6x3222x2x13x1. y032x1 x12 yln(2x3)x2 y01x12 yln(2x3)x2 yln(2x3)x2 32112 112. 12 f(x)x3bx2cxd P(0,2) M(1f(1))

6xy70.

(1) yf(x)

(2) yf(x)

(1) yf(x) P(0,2) d2 f(x)x3bx2cx2f(x)3x22bxc.

M(1f(1)) 6xy70

6f(1)70 f(1)1f(1)6.

32bc6

2bc3

1bc21 bc0

bc3. f(x)x33x23x2. (2)f(x)3x26x3. f(x)0

x1 2 x1 2

f(x)0 1 2x1 2.

f(x)x33x23x2 (1 2)(1 2)

(1 21 2)



13 f(x)mx3nx2(mnRm0) yf(x)(2f(2))

x

(1) m n

(2) f(x) (1) f(x)3mx22nx

f(2)03mn0 n3m. (2)n3mf(x)mx33mx2 f(x)3mx26mx. f(x)0 3mx26mx0

m0 x0 x2 f(x)(0)(2)

m0 0x2 f(x)(0,2)

m0 f(x)(0)(2)

m0 f(x)(0,2).