e a ax x x
f )()(2
++=
(1)当1=a 时,求)(x f 的单调递增区间; (2)若)(x f 的极大值是2
6-⋅e ,求a 的值.
3 .已知函数
1()ln(1),01x
f x ax x x
-=++
≥+,其中0a > ()I 若()f x 在x=1处取得极值,求a 的值; ()II 求()f x 的单调区间;
(Ⅲ)若()f x 的最小值为1,求a 的取值范围 .
4 .已知函数
()ln f x x x =.
(Ⅰ)求()f x 的单调区间;
(Ⅱ) 当1k ≤时,求证:()1f x kx ≥-恒成立.
5 .已知函数()ln a
f x x x
=-
,其中a ∈R . (Ⅰ)当2a =时,求函数()f x 的图象在点(1,(1))f 处的切线方程; (Ⅱ)如果对于任意(1,)x ∈+∞,都有()2f x x >-+,求a 的取值范围.
6 .已知函数2
()4ln f x ax x =-,a ∈R .
(Ⅰ)当1
2
a =
时,求曲线()y f x =在点(1,(1))f 处的切线方程; (Ⅱ)讨论()f x 的单调性.
7 .已知函数()e (1)x
f x x =+.
(Ⅰ)求曲线()y f x =在点(0,(0))f 处的切线方程;
(Ⅱ)若对于任意的(,0)x ∈-∞,都有()f x k >,求k 的取值范围.
8 .已知函数a ax x x f 23)(3
+-=,)(R a ∈.
(Ⅰ) 求)(x f 的单调区间;
(Ⅱ)曲线)(x f y =与x 轴有且只有一个公共点,求a 的取值范围.
9 .已知函数
22()2ln (0)f x x a x a =->.
(Ⅰ)若()f x 在1x =处取得极值,求实数a 的值; (Ⅱ)求函数()f x 的单调区间;
(Ⅲ)若()f x 在[1]e ,
上没有零点,求实数a 的取值范围.
10.已知曲线
()x f x ax e =-(0)a >.
(Ⅰ)求曲线在点(0,(0)f )处的切线;
(Ⅱ)若存在实数0x 使得0()0f x ≥,求a 的取值范围.
导数及其应用大题精选参考答案
1. 解: (1)f ′(x )=a -b
x 2,则有⎩
⎪⎨
⎪⎧
f 1=a +b +c =0f ′1=a -b =1,解得⎩
⎪⎨
⎪⎧
b =a -1,
c =1-2a .
(2)由(1)知,f (x )=ax +
a -1
x
+1-2a . 令g (x )=f (x )-ln x =ax +
a -1
x
+1-2a -ln x ,x ∈[1,+∞), 则g (1)=0,g ′(x )=a -
a -1x 2-1x =ax 2
-x -a -1x 2
=a x -1x -
1-a
a
x 2
,
(ⅰ)当0a
>1.
若1a
,则g ′(x )<0,g (x )是减函数,所以g (x )即f (x )≤1.
若x >1,则g ′(x )>0,g (x )是增函数,所以g (x )>g (1)=0,
即f (x )>ln x ,故当x ≥1时,f (x )≥ln x . 综上所述,所求a 的取值范围为[1
2
,+∞).
2.
3. (Ⅰ)
22222
'(),1(1)(1)(1)
a ax a f x ax x ax x +-=-=++++
∵()f x 在x=1处取得极值,∴2
'(1)0,120,f a a =+-= 即解得 1.a = (Ⅱ)
222
'(),(1)(1)
ax a f x ax x +-=++
∵0,0,x a
≥> ∴10.ax +>
①当2a ≥时,在区间(0,)'()0,f x +∞>上,∴()f x 的单调增区间为(0,).+∞ ②当02a <<时,
由'()0'()0f x x f x x >>
<<解得由解得
∴()f x +∞的单调减区间为(0).
(Ⅲ)当2a
≥时,由(Ⅱ)①知,()(0)1;f x f =的最小值为
当02a <<时,由(Ⅱ)②知,
()f x
在x =
(0)1,f f <= 综上可知,若()f x 得最小值为1,则a 的取值范围是[2,).+∞
4. 解: (Ⅰ) 定义域为()0,+∞ ,
'()ln 1f x x =+
令'()0f x =,得 1
e
x =
'()f x 与()f x 的情况如下:
所以()f x 的单调减区间为1(0,)e ,单调增区间为1(,)e
+∞