## 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.

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.

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

*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.

We have many more “use the Mean Value Theorem to prove (something or another)” problems. To access them, get your membership now — it’s quick and easy!

What are your questions about the Mean Value Theorem, or Rolle’s Theorem? Let us know below, and we’ll do our best to answer!

### Do you need immediate help with a particular textbook problem?

Head over to our partners at Chegg Study and gain (1) *immediate* access to step-by-step solutions to most textbook problems, probably including yours; (2) answers from a math expert about specific questions you have; AND (3) 30 minutes of free online tutoring. Please visit Chegg Study now.

If you use Chegg Study, we’d greatly appreciate hearing your super-quick feedback about your experience to make sure you’re getting the help
you need.

## What are your thoughts and questions?