Here graphs of three functions that are continuous within the closed interval zero to four.

Use the slider to vary the value of x within the interval. Notice how the function and derivative values vary with x.

On each curve, the function values at the endpoints are equal.

We see that in each case there is at least one point at which the derivative is equal to zero. In the first graph, this point is an absolute minimum value. In the second graph, it is an absolute maximum value. The function in the third graph is a constant, so the derivative is zero at all points in the interval.

These graphs illustrate Rolle's theorem. It states that if a function is continuous within the closed interval [a, b] and differentiable within the open interval (a, b), and if the function values at the endpoints are equal, then there is some number c in the open interval (a, b) such that the derivative at c is equal to zero.

Consider this graph. The function is not differentiable at points A and B, both of which have function values equal to zero. However, the function is continuous within the closed interval [a, b] and is differentiable within the open interval (a, b). According to Rolle's theorem, there is a point in the open interval where the derivative is zero.

Let's look at another function. Rolle's theorem does not apply to this function for the closed interval [a, b].

Such a point does exist at c, where we see that a line tangent to the curve is horizontal, and therefore has a slope of zero. Rolle's theorem is valid in this case.

The three graphed functions are all differentiable at their endpoints. However, this is not a necessary condition of Rolle's theorem.

What can we say about the values at the endpoints of the three functions? Click the "Submit" button after selecting your answer.

What can we say about the derivative values of each function? Click the "Submit" button after selecting your answer.

Why doesn't Rolle's theorem apply to this function for the closed interval [a, b]? Click the "Submit" button after selecting your answer.

Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education