📝 题目
例 1 设 $f\left( x\right)$ 为凹函数,且二次可导. 求证: $F\left( x\right) = {\mathrm{e}}^{f\left( x\right) }$ 也是凹函数.
💡 答案与解析
证 因为 $f\left( x\right)$ 为凹函数,所以 ${f}^{\prime \prime }\left( x\right) \geq 0$ . 又
$$ {F}^{\prime }\left( x\right) = {f}^{\prime }\left( x\right) {\mathrm{e}}^{f\left( x\right) },\;{F}^{\prime \prime }\left( x\right) = {f}^{\prime \prime }\left( x\right) {\mathrm{e}}^{f\left( x\right) } + {\left( {f}^{\prime }\left( x\right) \right) }^{2}{\mathrm{e}}^{f\left( x\right) } \geq 0, $$
即得 $F\left( x\right)$ 为凹函数.