Mean Value Theorem
The Mean Value Theorem states that if 
are real numbers and the function ![$f:[a,b] \to \mathbb{R}$](http://latex.artofproblemsolving.com/f/d/b/fdb437e0031d3b146dc3c59922d4a94e15a7ec29.png)
is differentiable on the interval 
, then there exists a value 
in such that
![\[f(c)=\dfrac{1}{b-a}\int_{a}^{b}f(x)dx.\]](http://latex.artofproblemsolving.com/1/a/8/1a8347c593c9df7eca6e55210e2b2e5a448bdff9.png)
In words, there is a number in such that 
equals the average value of the function in the interval ![$[a,b]$](http://latex.artofproblemsolving.com/8/e/c/8ecbd1ba3da8f2adef66a63f2ab32c47e63fa734.png)
.
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Proof
Other
Rolle's Theorem is a sub-case of this theorem. It states that if 
for two real numbers a and b, then there is a real number c such that 
and 
.
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