From a1216ac1465940fe09c99f8545b5ebf4620ff003 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?=E7=8E=8B=E8=BD=B2=E6=A5=A0?= <wkn339224@qq.com>
Date: Wed, 14 Jan 2026 13:55:37 +0800
Subject: [PATCH] vault backup: 2026-01-14 13:55:37

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+>[!note] 定理：
+>如果函数$f(x)$在区间$[a,b]$上可导，则其导函数$f'(x)$在$[a,b]$上有介值性质，即若$f(x)$在$[a,b]$上的值域为$[m,M]$,则$\forall \xi\in[m,M]$，总$\exists \eta\in[a,b],$有$\xi=f'(\eta)$.
+
+**证明**：若$m=M$，结论显然成立.若$m<M$,设$f'(x_1)=m,f'(x_2)=M$,不妨设$x_1<x_2$.任取$\xi\in(m,M)$,令$$g(x)=f(x)-\xi x,x\in[a,b].$$于是$g(x)$在$[a,b]$上可导，且$$g'(x_1)=f'(x_1)-\xi<0,g'(x_2)=f'(x_2)-\xi>0.$$由零值定理，$\exists  \eta\in(x_1,x_2) \subset(a,b),g'(\eta)=0\implies f'(\eta)=\xi$，证毕.
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