已知
f(
x)=
x3+
mx2-
x+2(
m∈
R)
如果函數(shù)的單調(diào)減區(qū)間恰為(-

,1),求函數(shù)
f(
x)的解析式;
(2)若
f(
x)的導(dǎo)函數(shù)為
f '(
x),對任意
x∈(0,+∞),不等式
f '(
x)≥2
xlnx-1恒成立,求實數(shù)
m的取值范圍.
(1)f(x)=x3-x2-x+2
(2)m的取值范圍是[ln2-ln3e,+∞).
(1)
f '(
x)=3
x2+2
mx-1,
由題意,
f '(
x)=3
x2+2
mx-1<0的解集是(-

,1),
即3
x2+2
mx-1=0的兩根分別為-

,1,將
x=1或-

代入方程3
x2+2
mx-1=0得
m=-1,
∴
f(
x)=
x3-
x2-
x+2,
(2)由題意知3
x2+2
mx-1≥2
xlnx-1在
x∈(0,+∞)恒成立,
即
m≥
lnx-
x在
x∈(0,+∞)恒成立,
設(shè)
h(
x)=
lnx-
x,則
h'(
x)=

-

,
令
h'(
x)=0得
x=

,
當(dāng)0<
x<

時,
h'(
x)>0;當(dāng)
x>

時,
h'(
x)<0,
∴當(dāng)
x=

時,
h(
x)取得最大值為
ln
-1=
ln2-
ln3
e,
表明
m≥
ln2-
ln3
e,
因此
m的取值范圍是[
ln2-
ln3
e,+∞).
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有極值.
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
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
在
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