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ROLLES THEOREM AND THE EXTREME VALUE THEOREM
Section 3.2
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When you are done with your homework, you should be able to…
Understand and use Rolles Theorem Understand and use the Mean Value Theorem
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ROLLE’S THEOREM: AN ILLUSTRATION
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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
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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
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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
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Can Rolle’s Theorem be applied to the function shown below on the interval ?
Yes No
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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
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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
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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.
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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.
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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
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