Explanation Given information: The given function f is, f ( x ) = 3 x 4 − 6 x 3 + x − 8 . Definition used: The function f ( x ) is concave upward or concave downward, (a) If f ″ ( x ) > 0 for every value of x in ( a , b ) , then the graph of function is concave upward on ( a , b ) . (b) If f ″ ( x ) < 0 for every value of x in ( a , b ) , then the graph of function is concave downward on ( a , b ) . Calculation: The function of f is, f ( x ) = 3 x 4 − 6 x 3 + x − 8 . Differentiate the above function with respect to x , f ′ ( x ) = d d x ( 3 x 4 − 6 x 3 + x − 8 ) = d d x ( 3 x 4 ) − d d x ( 6 x 3 ) + d d x ( x ) − d d x ( 8 ) [ d d x ( x n ) = n ⋅ x n − 1 ] = 12 x 3 − 18 x 2 + 1 Again, differentiate the above derivative with respect to x , f ″ ( x ) = d d x ( 12 x 3 − 18 x 2 + 1 ) = d d x ( 12 x 3 ) − d d x ( 18 x 3 ) + d d x ( 1 ) [ d d x ( x n ) = n ⋅ x n − 1 ] = 36 x 2 − 36 x Since, f ″ ( x ) is defined on everywhere

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

10th Edition

ISBN: 9781305657861

Author: Soo T. Tan

Publisher: Cengage Learning

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Chapter 4.2, Problem 34E

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The intervals where the graph of the function is concave upward and concave downward.

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Chapter 4.2, Problem 33E

Chapter 4.2, Problem 35E

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(b) Find the (instantaneous) rate of change of y at x = 5. In the previous part, we found the average rate of change for several intervals of decreasing size starting at x = 5. The instantaneous rate of change of fat x = 5 is the limit of the average rate of change over the interval [x, x + h] as h approaches 0. This is given by the derivative in the following limit. lim h→0 - f(x + h) − f(x) h The first step to find this limit is to compute f(x + h). Recall that this means replacing the input variable x with the expression x + h in the rule defining f. f(x + h) = (x + h)² - 5(x+ h) = 2xh+h2_ x² + 2xh + h² 5✔ - 5 )x - 5h Step 4 - The second step for finding the derivative of fat x is to find the difference f(x + h) − f(x). - f(x + h) f(x) = = (x² x² + 2xh + h² - ])- = 2x + h² - 5h ])x-5h) - (x² - 5x) = ]) (2x + h - 5) Macbook Pro

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Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

Ch. 4.1 - Explain each of the following: a. f is increasing...Ch. 4.1 - Prob. 2CQ Ch. 4.1 - Prob. 3CQ Ch. 4.1 - Prob. 4CQ Ch. 4.1 - Prob. 5CQ Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E

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The intervals where the graph of the function is concave upward and concave downward.

Solution Summary: The function f(x) is concave upward and downward.

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