题目
题型:陕西省模拟题难度:来源:
(Ⅰ)要使y=f(x)在(0,1)上单调递增,求a的取值范围;
(Ⅱ)当a>0时,若函数满足y极小值=1,y极大值=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004058-10276.png)
(Ⅲ)若x∈[0,1]时,y=f(x)图象上任意一点处的切线倾斜角为θ,求当0≤θ≤
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004059-30809.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004059-20827.png)
要使f(x)在(0,1)上单调递增,
则x∈(0,1)时,f′(x)≥0恒成立,
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004059-20468.png)
即当x∈(0,1)时,
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004059-24593.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004059-13564.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004100-96413.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004100-85990.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004100-68186.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004100-95163.png)
(Ⅱ)由
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004101-15923.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004101-67841.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004101-77486.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004101-15582.png)
∵a>0,∴当x变化时,f′(x)、f(x)的变化情况如下表:
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004102-78566.png)
∴y极小值=f(0)=b=1,y极大值=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004102-87382.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004102-26135.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004102-44652.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004103-96358.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004103-13838.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004103-49840.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004103-49615.png)
∴b=1,a=1,
故f(x)=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004103-50220.png)
(Ⅲ)当x∈[0,1]时,tanθ=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004104-82642.png)
由θ∈[0,
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004104-35733.png)
即x∈[0,1]时,0≤
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004104-65364.png)
当x=0时,a∈R,
当x∈(0,1]时,由
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004104-82501.png)
由(Ⅰ)知
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004105-20359.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004105-46710.png)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004105-79835.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004105-37132.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004106-46987.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004106-10645.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004106-47960.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004106-57456.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004106-25238.png)
综上,
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004107-51524.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004107-53869.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004107-11228.png)
核心考点
试题【函数y=f (x )=-x3+ax2+b(a,b∈R ),(Ⅰ)要使y=f(x)在(0,1)上单调递增,求a的取值范围;(Ⅱ)当a>0时,若函数满足y极小值=1】;主要考察你对函数的单调性与导数等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004048-38907.gif)
(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。
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004031-87195.png)
(Ⅰ)若f(x)在[1,+∞)上单调递增,求a的取值范围;
(Ⅱ)若定义在区间D上的函数y=f(x)对于区间D上的任意两个值x1、x2总有以下不等式
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004031-36401.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004023-87125.png)
(Ⅰ)若函数f(x)在区间(2m-1,m+1)上单调递减,求实数m的取值范围;
(Ⅱ)设函数
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004023-58840.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004012-13228.png)
(2)已知函数f(x)=ln(1+x)-
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004013-62675.png)
(3)若关于x的不等式
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004013-93306.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019004003-84783.png)
(1)求y=f(x)在点P(0,1)处的切线方程;
(2)设g(x)=f(x)+x-1仅有一个零点,求实数m的值;
(3)试探究函数f(x)是否存在单调递减区间?若有,设其单调区间为[t,s] ,试求s-t的取值范围?若没有,请说明理由。
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