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What is mean value theorem
The Mean Value Theorem is one of the most important theoretical tools in Calculus. It states that if f(x) is defined and continuous on the interval [a,b] and differentiable on (a,b), then there is at least one number c in the interval (a,b) (that is a < c < b) such that 



The special case, when f(a) = f(b) is known as Rolle's Theorem. In this case, we have f '(c) =0. In other words, there exists a point in the interval (a,b) which has a horizontal tangent. In fact, the Mean Value Theorem can be stated also in terms of slopes. Indeed, the number 
 
is the slope of the line passing through (a,f(a)) and (b,f(b)). So the conclusion of the Mean Value Theorem states that there exists a point $c\in(a,b)$ such that the tangent line is parallel to the line passing through (a,f(a)) and (b,f(b)).


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