Explanation Given: The function is f ( x ) = x 8 3 + x 5 3 . Calculation: Evaluate f ′ ( x ) as follows. f ′ ( x ) = d d x ( x 8 3 + x 5 3 ) = ( 8 3 ) x 5 3 + ( 5 3 ) x 2 3 Thus, f ′ ( x ) = ( 8 3 ) x 5 3 + ( 5 3 ) x 2 3 . Equate f ′ ( x ) to 0 to find the critical number. ( 8 3 ) x 5 3 + ( 5 3 ) x 2 3 = 0 x 2 3 3 ( 8 x + 5 ) = 0 x 2 3 = 0 or 8 x + 5 = 0 x = 0 or x = − 5 8 The critical numbers are x = 0 and x = − 5 8 . The second derivative f ″ ( x ) can be obtained from f ′ ( x ) = ( 8 3 ) x 5 3 + ( 5 3 ) x 2 3 as follows. f ′ ′ ( x ) = d d x ( ( 8 3 ) x 5 3 + ( 5 3 ) x 2 3 ) = 8 3 ( 5 3 ) x 2 3 + 5 3 ( 2 3 ) x − 1 3 = 40 9 x 2 3 + 10 9 x − 1 3 Thus, the value of f ″ ( x ) is f ′ ′ ( x ) = 40 9 x 2 3 + 10 9 x − 1 3 . Substitute x = − 5 8 in f ′ ′ ( x ) = 40 9 x 2 3 + 10 9 x − 1 3 . f ′ ′ ( − 5 8 ) = 40 9 ( − 5 8 ) 2 3 + 10 9 ( − 5 8 ) − 1 3 = ( 4

Finite Mathematics and Calculus with Applications

1st Edition

ISBN: 9781323188361

Author: Margaret Lial

Publisher: Pearson Education

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Chapter 13.3, Problem 66E

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To find: The critical numbers for the given function and corresponding relative maxima or minima using the suitable derivative test.

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Chapter 13.3, Problem 65E

Chapter 13.3, Problem 67E

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Consider the function f(x) = x²-1. (a) Find the instantaneous rate of change of f(x) at x=1 using the definition of the derivative. Show all your steps clearly. (b) Sketch the graph of f(x) around x = 1. Draw the secant line passing through the points on the graph where x 1 and x-> 1+h (for a small positive value of h, illustrate conceptually). Then, draw the tangent line to the graph at x=1. Explain how the slope of the tangent line relates to the value you found in part (a). (c) In a few sentences, explain what the instantaneous rate of change of f(x) at x = 1 represents in the context of the graph of f(x). How does the rate of change of this function vary at different points?

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Finite Mathematics and Calculus with Applications

Ch. 13.1 - Find where the function is increasing and...Ch. 13.1 - Prob. 2YT Ch. 13.1 - Prob. 3YT Ch. 13.1 - Prob. 4YT Ch. 13.1 - Prob. 1WE Ch. 13.1 - Prob. 2WE Ch. 13.1 - Prob. 3WE Ch. 13.1 - Prob. 4WE Ch. 13.1 - Prob. 5WE Ch. 13.1 - Prob. 6WE

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The critical numbers for the given function and corresponding relative maxima or minima using the suitable derivative test.

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To find: The critical numbers for the given function and corresponding relative maxima or minima using the suitable derivative test.

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Chapter 13 Solutions