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Does the mean value theorem also apply to lateral derivatives?

Let $f: [a,b] \longrightarrow \mathbb{R}$ continuous and $f$ right differentiable in $(a,b)$, then, theres exists $c \in (a,b)$ such that $$f'_{+}(c) = \frac{f(b)-f(a)}{b-a}$$

In the proof of the Means Value Theorem, based on Rolle's Theorem, it wasn't clear to me what would happen if we changed this hypothesis

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Nope, here is a counter example:

enter image description here $f(x) = |x|$

Take $a=-1$, $b=1$, but there is no right tangent with slope $f'_{+}(x)=0$.

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Consider the function f(x) = $ 2-|x|$ on $[-2,2].$

It is continuous and right differentiable but there is no point in $(-2,2)$ with $f'_{+}(c) = \frac{f(b)-f(a)}{b-a}=0 $$

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