Mean Value Theorem

The Mean Value Theorem states that if $a < b$ are real numbers and the function $f:[a,b] \to \mathbb{R}$ is differentiable on the interval $(a,b)$, then there exists a value $c$ in $(a,b)$ such that

\[f(c)=\dfrac{1}{b-a}\int_{a}^{b}f(x)dx.\]

In words, there is a number $c$ in $(a,b)$ such that $f(c)$ equals the average value of the function in the interval $[a,b]$.

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Proof

Other

Rolle's Theorem is a sub-case of this theorem. It states that if $f(a)=f(b)=0$ for two real numbers a and b, then there is a real number c such that $a<c<b$ and $f'(c)=0$.

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