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 ) = − 2 3 x 3 − 1 2 x 2 + 3 x − 4 . Obtain the derivative of the function as follows. f ′ ( x ) = ( − 2 3 x 3 − 1 2 x 2 + 3 x − 4 ) ′ = − 2 3 ( 3 x 2 ) − 1 2 ( 2 x ) + 3 ( 1 ) − 0 = − 2 x 2 − 1 x + 3 Set f ′ ( c ) = 0 and find the value of c . − 2 c 2 − 1 c + 3 = 0 ( 2 c 2 + 1 c − 3 ) = 0 2 c 2 + 3 c − 2 c − 3 = 0 c ( 2 c + 3 ) − 1 ( 2 c + 3 ) = 0 Simplify further as follows. c ( 2 c + 3 ) − 1 ( 2 c + 3 ) = 0 ( c − 1 ) ( 2 c + 3 ) = 0 c − 1 = 0 or 2 c + 3 = 0 c = 1 or c = − 3 2 Therefore, the critical numbers are c = 1 and c = − 3 2 . These points determine the intervals ( − ∞ , − 3 2 ) , ( − 3 2 , 1 ) and ( 1 , ∞ ) . Consider the points from each of these intervals and find the sign of the derivative function at these intervals. Consider a point x = − 2 in the interval ( − ∞ , − 3 2 ) and find the sign of the derivative at this point. f ′ ( − 2 ) = − 2 ( − 2 ) 2 − 1 ( − 2 ) + 3 = − 2 × 4 + 2 + 3 = − 8 + 5 = − 3 Consider a point x = 0 in the interval ( − 3 2 , 1 ) and find the sign of the derivative at this point

Finite Mathematics and Calculus with Applications

1st Edition

ISBN: 9781323188361

Author: Margaret Lial

Publisher: Pearson Education

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Chapter 13.2, Problem 18E

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To find: The x-values of all points where the function f(x)=−23x3−12x2+3x−4 has relative extrema and to find the value of relative extrema.

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Chapter 13.2, Problem 17E

Chapter 13.2, Problem 19E

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Chapter 13 Solutions

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 x -values of all points where the function f ( x ) = − 2 3 x 3 − 1 2 x 2 + 3 x − 4 has relative extrema and to find the value of relative extrema.

Solution Summary: The author explains how to find the x-values of all points where the function f(x)=-23x

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To find: The x-values of all points where the function f(x)=−23x3−12x2+3x−4 has relative extrema and to find the value of relative extrema.

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Chapter 13 Solutions