Explanation Definition used: First derivative test: Suppose c to be a critical number for a function f . Assume that f is continuous on ( a , b ) and differentiable on ( a , b ) except at c . The interval ( a , b ) should have only one critical number of f . Relative maximum : A function is said to have a relative maximum f ( c ) at c if f ′ ( x ) > 0 in the interval ( a , c ) and f ′ ( x ) < 0 in the interval ( c , b ) . Relative minimum: A function is said to have a relative maximum f ( c ) at c if f ′ ( x ) < 0 in the interval ( a , c ) and f ′ ( x ) > 0 in the interval ( c , b ) . Calculation: The given function is f ( x ) = − 4 3 x 3 − 21 2 x 2 − 5 x + 8 . Obtain the derivative of the function as follows. f ′ ( x ) = ( − 4 3 x 3 − 21 2 x 2 − 5 x + 8 ) ′ = − 4 3 ( 3 x 2 ) − 21 2 ( 2 x ) − 5 ( 1 ) + 0 = − 4 x 2 − 21 x − 5 Set f ′ ( c ) = 0 and find the value of c . − 4 c 2 − 21 c − 5 = 0 ( 4 c 2 + 21 c + 5 ) = 0 4 c 2 + 20 c + c + 5 = 0 4 c ( c + 5 ) + 1 ( c + 5 ) = 0 Simplify further as follows. 4 c ( c + 5 ) + 1 ( c + 5 ) = 0 ( c + 5 ) ( 4 c + 1 ) = 0 c + 5 = 0 or 4 c + 1 = 0 c = − 5 or c = − 1 4 Therefore, the critical numbers are c = − 5 and c = − 1 4 . These points determine the intervals ( − ∞ , − 5 ) , ( − 5 , − 1 4 ) and ( − 1 4 , ∞ ) . Consider the points from each of these intervals and find the sign of the derivative function at these intervals. Consider a point x = − 6 in the interval ( − ∞ , − 5 ) and find the sign of the derivative at this point. f ′ ( − 6 ) = − 4 ( − 6 ) 2 − 21 ( − 6 ) − 5 = − 4 × 36 + 126 − 5 = − 144 + 126 − 5 = − 23 Consider a point x = − 1 in the interval ( − 5 , − 1 4 ) and find the sign of the derivative at this point

Finite Mathematics and Calculus with Applications (10th Edition)

10th Edition

ISBN: 9780321979407

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Chapter 13.2, Problem 17E

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To find: The x-values of all points where the function f(x)=−43x3−212x2−5x+8 has relative extrema and the value of relative extrema.

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Chapter 13.2, Problem 16E

Chapter 13.2, Problem 18E

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Finite Mathematics and Calculus with Applications (10th Edition)

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 x -values of all points where the function f ( x ) = − 4 3 x 3 − 21 2 x 2 − 5 x + 8 has relative extrema and the value of relative extrema.

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To find: The x-values of all points where the function f(x)=−43x3−212x2−5x+8 has relative extrema and the value of relative extrema.

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