题目
题型:河北省月考题难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019060806-85455.png)
(1)试求实数a的取值范围.
(2)若a=2,求f(x)=c有三个不同实根时,c的取值范围.
(说明:第二问能用f(x)表达即可,不必算出最结果.)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019060806-45899.png)
∴f′(x)=x2﹣ax+a2﹣13,
∴f(x)在区间(1,4)内为减函数,在区间(6,+∞)上为增函数.
∴f′(x)=x2﹣ax+a2﹣13≤0在区间(1,4)上恒成立,
f′(x)=x2﹣ax+a2﹣13≥0在区间(1,4)上恒成立,
由f′(x)=x2﹣ax+a2﹣13开口向上,
∴只需
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019060806-10529.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019060807-29579.png)
∴a的取值范围为[1,3].
(2)∵a=2,f(x)=
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019060807-40831.png)
∴f′(x)=x2﹣2x﹣9,
∴令f′(x)=x2﹣2x﹣9≥0即x≤1﹣
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019060807-13604.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019060807-54350.png)
∴f(x)的增区间为(﹣∞,1﹣
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019060808-94833.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019060808-84801.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019060808-77857.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019060808-95313.png)