Extended Mean Value Theorem

Theorem

Let $f$ be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.

Let $f'$ be the derivative of $f$.

Let $f'$ also be a real function which is continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.

Then:

$\exists \xi \in \openint a b: \map f b = \map f a + \paren {b - a} \map {f'} a + \dfrac 1 {2!} \paren {b - a}^2 \map {f' '} \xi$

Proof

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