Mean value theorem

The mean value theorem simply states that for suitably well-behaved functions there is always at least one point in an interval where the instantaneous derivative equals the average derivative for the whole interval. Suppose $𝑓 : [𝑎, 𝑏] \to ℝ$ is continuous and is $𝐶 1$ differentiable on $(𝑎, 𝑏)$ . Then there exists a $𝑐 \in (𝑎, 𝑏)$ such that #m/thm/anal

𝑓' (𝑐) = 𝑓 (𝑏) - 𝑓 (𝑎) 𝑏 - 𝑎

Proof

Let

𝑟 = 𝑓 (𝑏) - 𝑓 (𝑎) 𝑏 - 𝑎

and define $𝑔 (𝑥) = 𝑓 (𝑥) - 𝑟 𝑥$ , which is clearly $𝐶 1$ differentiable. Since $𝑔 (𝑎) = 𝑔 (𝑏)$ , it follows from Rolle's theorem that there exists a $𝑐 \in (𝑎, 𝑏)$ such that $𝑔' (𝑐) = 0$ , i.e. $𝑓' (𝑐) = 𝑟$ , as required.

This is a simple generalization of Rolle's theorem for differentiable functions.

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