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Theorem 6.5.9: Mean Value Theorem

If f is continuous on [a, b] and differentiable on (a, b), then there exists a number c in (a, b) such that
f'(c) =
If f and g are continuous on [a, b] and differentiable on (a, b) and g'(x) # 0 in (a, b) then there exists a number c in (a, b) such that

Proof:

The first version of the Mean Value theorem is actually Rolle's theorem in disguise. A simple linear function can convert one situation into the other:

geometric interpretation of MVT

We need a linear function (linear so that we can easily compute its derivative) that maps the line through the two points ( a, f(a) ) and ( b, f(b) ) to the points ( a, 0 ) and ( b, 0 ). If we subtract that map from the function we will be in a situation where we can apply Rolle's theorem.

To find the equation of such a line is easy:

Thus, we define the following function:

Then h is differentiable in ( a, b ) with h(a) = h(b) = 0. Therefore, Rolle's theorem guaranties a number c between a and b such that h'(c) = 0. But then

which is exactly what we had to show for the first part.

The second part is very similar. You can fill in the details yourself by considering the function

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