Mean Value Theorem

If a function is:

Continuous on [a, b].

Differentiable on (a, b).

Then, there exists a point, c pertenece (a, b) such that:

Mean Value Theorem

Mean Value Theorem Interpretation

The geometric interpretation of the Mean Value Theorem tells us that there is a point where the tangent is parallel to the secant.

Rolle's Theorem is a special case of the Mean Value Theorem, in that f(a) = f(b).

Example

Can the Mean Value Theorem be applied to f(x) = x3 on [−1, 2]?

The function f(x) is continuous on [−1, 2] and differentiable on (−1, 2) so the Mean Value Theorem can be applied:

Mean Value Theorem

Mean Value Theorem Solution