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 , and c is the only critical number of f in ( a , b ) . f ( c ) is a relative maximum of f at c if f ′ ( x ) > 0 in the interval ( a , c ) and f ′ ( x ) < 0 in the interval ( c , b ) . f ( c ) is a relative minimum f at c if f ′ ( x ) < 0 in the interval ( a , c ) and f ′ ( x ) > 0 in the interval ( c , b ) . 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 ) = 2 x 3 + 3 x 2 − 12 x + 5 . Obtain the derivative of the function as follows. f ′ ( x ) = ( 2 x 3 + 3 x 2 − 12 x + 5 ) ′ = 2 ( 3 x 2 ) + 3 ( 2 x ) − 12 = 6 x 2 + 6 x − 12 Set f ′ ( c ) = 0 and find the value of c . 6 c 2 + 6 c − 12 = 0 c 2 + c − 2 = 0 c 2 + 2 c − c − 2 = 0 c ( c + 2 ) − 1 ( c + 2 ) = 0 Simplify further as follows. c ( c + 2 ) − 1 ( c + 2 ) = 0 ( c − 1 ) ( c + 2 ) = 0 c − 1 = 0 or c + 2 = 0 c = 1 or c = − 2 Therefore, the critical numbersare c = − 2 and c = 1 . These points determine the intervals ( − ∞ , − 2 ) , ( − 2 , 1 ) and ( 1 , ∞ )

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, Problem 30RE

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To find: The x-values of all points where the function f(x)=2x3+3x2−12x+5 has relative maxima and minima, and to find the values of all relative extrema.

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Chapter 13, Problem 29RE

Chapter 13, Problem 31RE

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Chapter 13 Solutions

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 ) = 2 x 3 + 3 x 2 − 12 x + 5 has relative maxima and minima , and to find the values of all relative extrema.

Solution Summary: The author explains how to find the x -values of all points where the function f(x)=2x,3+3

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To find: The x-values of all points where the function f(x)=2x3+3x2−12x+5 has relative maxima and minima, and to find the values of all relative extrema.

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Chapter 13 Solutions