## Mean Value Theorem & Rolle’s Theorem

### Mean Value Theorem & Rolle’s Theorem: Problems and Solutions

Are you trying to use the Mean Value Theorem or Rolle’s Theorem in Calculus? Let’s introduce the key ideas and then examine some typical problems step-by-step so you can learn to solve them routinely for yourself.

CALCULUS SUMMARY: Mean Value Theorem & Rolle's Theorem

Problem #1: Straightforward Application of Rolle's Theorem

Consider the function $f(x) = 9 – (x-3)^2$ on the interval $[0, 6]$. (Note that $f(0) = f(6) = 0$.) Find the value(s) of $c$ that satisfy Rolle’s Theorem.

Problem #2: Straightforward Application of the Mean Value Theorem

Consider the function $f(x) = x^2$ on the interval $[1, 4]$. Find the value(s) of $c$ that satisfy the Mean-Value Theorem.

Problem #3: Prove if

*f '(x)*> 0, then*f(x)*is an increasing function Use the Mean Value Theorem to prove that if $f(x)$ is differentiable and $f'(x) > 0$ for all $x$, then $f(x)$ is an increasing function.

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