解:(1)由 得![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019062339-28662.gif) ①曲线y=f(x)在点P(1,f(1))处的切线的斜率k=f"(1)=2; ②f"(x)=x+x2=x(x+1) 由f"(x)<0,得-1<x<0, 由f"(x)>0得x<-1或x>0 由题意,![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019062339-38159.gif) 即![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019062339-31638.gif) 解得![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019062339-24430.gif) 故m的取值范围为 。 (2)∵![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019062340-46086.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019062340-71403.gif) ∴![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019062340-25807.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019062340-85816.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019062341-61914.gif) ∴![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019062341-25110.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019062341-53986.gif) ∴![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019062341-98828.gif) 要证Tn≤f"(1) -1,其中n∈N* 即证![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019062342-75709.gif) 当n=1时,T1=1,f"(1)-1=1, 此时,Tn=f"(1)-1成立, 当n=2时,![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019062342-71589.gif) 不等式Tn<f"(1)-1成立 当n≥3时,![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019062342-10356.gif) ![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019062342-13229.gif) 而![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019062342-44533.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019062343-59452.gif) ∵![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019062343-68532.gif) ∴![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019062343-22001.gif) ∴当n≥3时,![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019062343-91317.gif) ![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019062344-44800.gif) 即![](http://img.shitiku.com.cn/uploads/allimg/20191019/20191019062344-51354.gif) ∴当n≥3时,不等式 也成立, 综上所述,对任意的n∈N*,不等式Tn≤f"(1)-1成立。 |