利用导数解决双变量之极值点偏移

aa利用导数决双变量之值点偏移

f(x)

112

e

x

f(x

f()f(xx)122

1

fxf

yf

x2

x2

xx212

1天

f(x)xe

求

f(x

yg()

yf(x)

x

f()g(x)

xx,(x)f(x),xx211212

kfx)xln（Rxx

(1

,0)且21

2

k

12

2e2

f

x

(R

f(x

x,x1

x

f)lnx

1x

g(x)ax(x)f(x)(x(0,1(x)f(xlnaxf(x))(x,)(,y)e

3

f(x)x

2

)xlnx

f(x

x

f()f(a)

,1

2

f(x

f'(

x

2)

f)xlnx

a2

(a

x1

x1

e1

4ff

x,x1

f

x12

a

f

f

1

f(aln

2

yf(x)

[

()()axy()

a）hx)()xx,0)B(x,0)12xh'()h(x))25

f)xmx(m为常数()

m

gx)2fx

2

x)

h(cx

2

bx

yx)h'(

x2)

已知函数f2x,f()x立，0，求f()在g

x

f

x

x2

11

.6f)xxg()x

(xf()(x)F()(1,a,}a,bh)min{fx),g()}xh(x)cx(x)F()xf()ln

x

2

(x

f(x

x,12

f(x)f()1

7xxx12xxx12f'()

)e))ex2xex2x2

.2

2

当fx)0,y(x)单调递;x[0x)0,f(x)单调递

(x-调递增；递减

xef(x)f(xe)ex]1x

令g(x))

2x

xx))e

2x

令())

2x

h))

2

2

0,)在递减()x)在递g()g(0)ey[(1)e

x

]在0调减，但x时yf()(0f()f)所以，f(x)f(xx时x12122axg]ggffff(xfxf(2x((x13.（）k,0)2e11（）知0xxx)()(),x即可证；e

()(1)e

x=1

]

1e

e

x

F(

x2)e

x

'()e

2x

e

从

2x-2

0,所F

F8极大极大121211222极大极大121211222

-1，所以x>1时，有

x由（)则xxx盾。121212(x1)(（f(x),xxxx盾。121212(妨xx121)>))f(2-x)f(2-x)22222f(x)f(2-x)，12xx112f(x的(0,

f

x

g(xax

g)

1

g

gx)(0,

g)

g

g

g)

(0,)

(

1())lna

因

0(g((xln0e

xlnxax112xln1xln1()xx21x121xlnxxaxln1x2x2t1lntxt212gtlntttt

函数

g

g

不等式t

2t

立，

所证不等式

1

2

立．

1h(x)f()(x)x)()xx

h

1.(x)x92233x12222233x1222

(0,

11(0,ax

1x

min

H()

111)x4

x

x

xH(设)00

112axbxx00

1111aln)lnxx1(xx)(1,x(0,1))

xx)a证

x1xxln(x)1()22xlnx1ln1)1xx1ln2xxx)1)(x)xxx122(x)xln()2ln2xt2xx211

2(t()t(Ft

tt(

Ft)t

tt

F(tlnt

tt

(1)

lnt

tt

ln

)2

)

x)x22ln2x11

.ln(x)

2(xx

4xx)x2

4

lnx12

12

12

12

G(x)ln

x

G(xlnx(0,x2x2

220.83e

G(xx1212

1

lne

e

xe

e

10f(x)

(0,

x

(0,

g()

lnx

g

x

0

x

0

(x)

(0,)

(

g)

g()最大

1e

()

x)xg(x)0

g()

(x

lnx

(0,

2

e1

x

1）

xx1

112122000max02122000max0,ln1221a()0,0xx1221等x121xlnax,lna(xa2xx2xlnxxx2(1x0ln1122(1t(0,1)lntt(0,1)t2ht)ln

t

(t(t(

)

(0,1)h

ht)

t(0,1)

h(1)

ht)

t(0,1)

t

)h

(

,1)h

ht)(0,

)

t

,1)

(1)

t)

t(0,1)

x2axf

2f221=00f

2x=lnxflnx2x2200020

0=2xxx0fxlnx

xax0

x2a

lnxx

xxxggx

12x212122lnf2212x221(x212122lnf2212x221()219x

x

x

x

ln

x

g

x

g

min

x,x1

f

min

x12lnax0,ln2ax0112221xx1xln212xx1=x1xxxxx2122xxxxx1111x2htlntxtttthtln1tln1

2121

f2=f

yf2

f(xf2ln－()xaxg

ax

(g

－－=2()，)x2a－时

－=

xa，)2

=时()x

2x

m(xt<t1mfxt<t1mfxmxx

xxmxxxmx

2(lnxx(x2(lnlnxmxx2(lnxxh))(2(lnxlnx)(2x1)(xx)0lnxxxx1()tut≥t2

1t

1t

<ulnt

1

f

1mxxx

x

m

x

1xmm

f

f

mx

1m

f

f

m

f=fx1mxfmmm时(x)f()2mx

g

x

m

32

xmxx12

x

x

h()ln

lnx

lnx

14xx111t2xx111t2

12

12

()2

h

1

cxy=

()[2

x

(x)]2xln1x)x(x)[x)2(x)1x11202x2121xt11513tm2t22

2

1212

122

tt

t

tt(t

1]2

ln

y)h2)

（1）x>0,恒f()

alnax2x2

x>0成立H

ln2x,H（xxxx(0,),H/()0,

2H/)0,

H（x）12H(e)，所e2e22）函gg

a

g

axx

0

h

x

1时a

h

x

1a

a0

x，

12

lnax0,lnlnx2

11lnx2

121x2

221l2212xx112

2，lntttlntt，ttt2t1,调递减t122t15

112x2