In this applet you can control the location of the two endpoints, and . You can also control the position of the tangent line with the slider marked .
Set the two endpoints wherever you like, then move the tangent line until it is tangent to a point on the graph between the two endpoints. This point of tangency has -coordinate , one of the values guaranteed in the theorem. See if you can create an interval with only one value of , and see if you can create an interval with more than one value of in it.
The Mean Value Theorem:
If is continuous on the closed interval and differentiable on the open interval , then there exists a number in such that