随机刷题

设可导函数 $f\left(x\right)$ 严格单调递增且满足 ${\int }_{-1}^{1}f\left(x\right) \mathrm{d}x = 0$,记 $a = {\int }_{0}^{1}f\left(x\right) \mathrm{d}x$.
(1) 证明 $a > 0$；
(2) 令 $F\left(x\right) = a\left({1 - {x}^{2}}\right) + {\int }_{1}^{x}f\left(t\right) {dt}$,证明:存在 $\xi \in \left({-1,1}\right)$,使得 ${F}^{\prime \prime }\left(\xi \right) = 0$.