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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:

  1. f is continuous on the closed interval [a, b]
  2. 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?


Given f(x) = x3x, a = 0 and b = 2. Use the Mean Value Theorem to find c.


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) = 3x2 – 1

We get the equation

But c must lie in (0, 2) so


Intuition behind the Mean Value Theorem.

The Mean Value Theorem


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