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) Use the Intermediate Value Theorem to show that there is one such that f"(a) = 0. point x = a

) Use the Intermediate Value Theorem to show that there is one such that f"(a) = 0. point x = a

College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter3: Functions
Section3.3: More On Functions; Piecewise-defined Functions
Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...
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uploa... Sy 10- (a) Locate all the local minima of f(x). Briefly justify your answer. (b) State the interval(s) in which f is concave down. Briefly justify your answer. 3. Consider the function x5 + x? 20 f (x) = e-* - (a) Use the Intermediate Value Theorem to show that there is one point x = a such that f"(a) = 0. (b) Prove that there is only one point for which f"(a) = 0 [Hint: look at the third derivative]. Is this an inflection point? Explain. (c) Assume that f"(x) > 0 for x < a and f"(x) < 0 for x > a. What does the first derivative test (applied to f) say about f'(x)? (d) Use the result obtained in part (c) to determine the number of critical points of f. 1. A right circular cylinder of height h 2x is inscribed in a sphere of fixed radius R.
Transcribed Image Text:uploa... Sy 10- (a) Locate all the local minima of f(x). Briefly justify your answer. (b) State the interval(s) in which f is concave down. Briefly justify your answer. 3. Consider the function x5 + x? 20 f (x) = e-* - (a) Use the Intermediate Value Theorem to show that there is one point x = a such that f"(a) = 0. (b) Prove that there is only one point for which f"(a) = 0 [Hint: look at the third derivative]. Is this an inflection point? Explain. (c) Assume that f"(x) > 0 for x < a and f"(x) < 0 for x > a. What does the first derivative test (applied to f) say about f'(x)? (d) Use the result obtained in part (c) to determine the number of critical points of f. 1. A right circular cylinder of height h 2x is inscribed in a sphere of fixed radius R.
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