📝 题目
例 7 设 $f\left( x\right) \in C\left\lbrack {a,b}\right\rbrack ,{f}^{\prime }\left( a\right)$ 存在,并设 $\eta$ 满足
$$ {f}^{\prime }\left( a\right) > \eta > \frac{f\left( b\right) - f\left( a\right) }{b - a}. $$
求证: $\exists \xi \in \left( {a,b}\right)$ ,使得 $\frac{f\left( \xi \right) - f\left( a\right) }{\xi - a} = \eta$ .
💡 答案与解析
证 考虑函数 $g\left( x\right) = \left\{ \begin{array}{ll} \frac{f\left( x\right) - f\left( a\right) }{x - a}, & a < x \leq b, \\ {f}^{\prime }\left( a\right) , & x = a. \end{array}\right.$
容易验证 $g\left( x\right) \in C\left\lbrack {a,b}\right\rbrack$ ,且 $g\left( a\right) > \eta > g\left( b\right)$ . 由连续函数的介值定理, $\exists \xi \in \left( {a,b}\right)$ ,使得 $g\left( \xi \right) = \eta$ ,即 $\frac{f\left( \xi \right) - f\left( a\right) }{\xi - a} = \eta$ .