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 is continuous and is differentiable on . Then there exists a such that #m/thm/anal

Proof

Let

and define , which is clearly differentiable. Since , it follows from Rolle's theorem that there exists a such that , i.e. , as required.

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

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