Rolle's Theorem

In this applet you can control the line using the given slider. Notice that as long as is between the local maximum and the local minimum we get three intervals on which Rolle's Theorem guarantees there is at least one point where the derivative is zero: to , to , and to . Rolle's Theorem: Let be continuous on the closed interval and differentiable on the open interval . If then there is at least one number in such that .