2013-12-11: 设 $f:[0,1]\to[0,1]$ 是 $C^2$ 函数, $f(0)=f(1)=0$, 且 $f''(x)<0$, $\forall\ x\in[0,1]$. 记曲线 $\left\{(x,f(x));\ x\in [0,1]\right\}$ 的长度为 $L$. 证明: $L<3$.

证明:  由 Rolle 定理, \[ \exists\ \xi\in (0,1),\mathrm{\ s.t.\ } f'(\xi)=0. \] 又由 $f''<0$ 知[ f'(x)left{begin{array}{ll}>0,&0<x<xi,\<0,&xi<x<1.end{array}right. ] 于是begin{equation*} begin{aligned} L&=int_0^1sqrt{1+f'^2(x)}mathrm{,d}x\ &=int_0^xi +int_xi^1 sqrt{1+f'^2(x)}mathrm{,d}x\&<int_0^xi [1+f'(x)]mathrm{,d}x+int_{xi}^1 [1-f'(x)]mathrm{,d}x\&=xi+f(xi)-f(0)+(1-xi)-[f(1)-f(xi)]\ &=1+2f(xi)\ &leq 3. end{aligned} end{equation*}