Rolle's theorem states that if $f$ is a function that satisfies:

1.  f is continuous on the closed interval $\left[a\,b\right]$,

2.  f is differentiable on the open interval ($a\,b$), and

3.  $f\left(a\right)\=f\left(b\right)$

then there exists a point $c$ in the open interval ($a\,b$) such that f'($c$) = 0.

The routine RollesTheorem takes an expression representing the function, checks that the requirements of the theorem hold, and then plots the expression and all points where the derivative is zero.

The mean value theorem is a generalization of Rolle's theorem which states that if $f$ is a function that satisfies:

1.  f is continuous on the closed interval $\left[a\,b\right]$, and

2.  f is differentiable on the open interval ($a\,b$),

then there exists a point $c$ in the open interval ($a\,b$) such that f'($c$) = $\frac{f\left(b\right)-f\left(a\right)}{b-a}$ where the right-hand side is the slope of the line connecting the points ($a\,f\left(a\right)$) and ($b\,f\left(b\right)$).  The Mean Value Theorem can be derived from Rolle's Theorem by considering the function $g\left(x\right)\=f\left(x\right)-\frac{\left(f\left(b\right)-f\left(a\right)\right)\left(x-a\right)}{b-a}$.

The routine MeanValueTheorem takes an expression representing the function, checks that the requirements of the theorem hold, and then plots the expression and all points where the derivative equals the slope of the secant line connecting the end points of the graph of $f$ on $\left[a\,b\right]$.

You can also learn about the Mean Value Theorem using the MeanValueTheoremTutor command.