ROLLES THEOREM AND THE EXTREME VALUE THEOREM Section 3.2

When you are done with your homework, you should be able to… Understand and use Rolles Theorem Understand and use the Mean Value Theorem

ROLLE’S THEOREM: AN ILLUSTRATION

ROLLES THEOREM Let f be continuous on a closed interval and differentiable on the open interval . If then there is at least one number c in such that .

PROOF Let Case 1: If is a constant on the interval and Case 2: Suppose . By the Extreme Value Theorem, you know that f has a maximum at some c in the interval. Since , this maximum does not occur at either endpoint. So f has a maximum in the open interval . This implies that is a relative maximum and thus a critical number of f. Finally, because f is differentiable at c, you can conclude that

PROOF CONTINUED… Case 3: Suppose . By the Extreme Value Theorem, you know that f has a minimum at some c in the interval. Since , this minimum does not occur at either endpoint. So f has a minimum in the open interval . This implies that is a relative minimum and thus a critical number of f. Finally, because f is differentiable at c, you can conclude that

Can Rolle’s Theorem be applied to the function shown below on the interval ? Yes No

Show that Rolle’s Theorem be applied to the function shown below on the given interval. What is the exact value of c? 2.5 0.0

THE MEAN VALUE THEOREM If f is continuous on a closed interval and differentiable on the open interval then there exists at least one number c in the open interval such that

Determine whether the Mean Value Theorem can be applied to f on the closed interval If the MVT can be applied, find all values of c in the open interval such that The MVT can be applied. c = 1. The MVT can be applied. c = -1 or 1. The MVT cannot be applied since f is not continuous on the closed interval. The MVT cannot be applied since f is not differentiable on the open interval.

WORK IT OUT. a. Graph the function f on the. given interval. b WORK IT OUT!!! a. Graph the function f on the given interval. b. Find and graph the secant line through points on the graph of f at the endpoints of the given interval. c. Find and graph any tangent lines to the graph of f which are parallel to the secant line.

If the graph of a function has three x-intercepts, then it must have at least two points at which its tangent line is horizontal. True False