# Mean Value Theorem

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## Definition of the Mean Value Theorem

**What is the Mean Value Theorem?**
Let

*f* be a function that satisfies the following hypotheses:

*f* is continuous on the closed interval [*a*, *b*]
*f* is differentiable on the open interval (*a*, *b*)

Then there is a number *c* in (*a*, *b*) such that

**How to use the Mean Value Theorem?**
**Example: **

Given *f*(*x*) = *x*^{3} – *x*, *a* = 0 and *b* = 2. Use the Mean Value Theorem to find *c*.

**Solution: **

Since *f* is a polynomial, it is continuous and differentiable for all *x*, so it is certainly continuous on [0, 2] and differentiable on (0, 2).

By the Mean Value Theorem, there is a number *c* in (0, 2) such that

*f*(2) – *f*(0) = *f *’(*c*) (2 – 0)

We work out that* f*(2) = 6, *f*(0) = 0 and *f ‘*(*x*) = 3*x*^{2} – 1

We get the equation

But *c* must lie in (0, 2) so

## Video

Intuition behind the Mean Value Theorem.

The Mean Value Theorem

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