4.4 The Mean Value Theorem | Calculus Volume 1 - Lumen Learning

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Proof

Let [latex]k=f(a)=f(b)[/latex]. We consider three cases:

[latex]f(x)=k[/latex] for all [latex]x \in (a,b)[/latex].
There exists [latex]x \in (a,b)[/latex] such that [latex]f(x)>k[/latex].
There exists [latex]x \in (a,b)[/latex] such that [latex]f(x)k[/latex], the absolute maximum is greater than [latex]k[/latex]. Therefore, the absolute maximum does not occur at either endpoint. As a result, the absolute maximum must occur at an interior point [latex]c \in (a,b)[/latex]. Because [latex]f[/latex] has a maximum at an interior point [latex]c[/latex], and [latex]f[/latex] is differentiable at [latex]c[/latex], by Fermat’s theorem, [latex]f^{\prime}(c)=0[/latex].

Case 3: The case when there exists a point [latex]x \in (a,b)[/latex] such that [latex]f(x)