• Rolle’s Theorem
• Mean Value Theorem
• L'Hôpital's rule

If a function is:

Continuous on [a, b].

Differentiable on (a, b).

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

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:

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