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 ) = − 3 x 2 + 2 x − 5 . Obtain the derivative of the function as follows. f ′ ( x ) = ( − 3 x 2 + 2 x − 5 ) ′ = − 3 ( 2 x ) + 2 ( 1 ) − 0 = − 6 x + 2 Set f ′ ( c ) = 0 and find the value of c . − 6 c + 2 = 0 6 c = 2 c = 2 6 c = 1 3 Therefore, the critical number is c = 1 3

Finite Mathematics and Calculus with Applications

1st Edition

ISBN: 9781323188361

Author: Margaret Lial

Publisher: Pearson Education

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Chapter 13, Problem 28RE

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

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Chapter 13, Problem 27RE

Chapter 13, Problem 29RE

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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 ) = − 3 x 2 + 2 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)=-3x 2+2x-5 has relative maxima and minima

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

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Chapter 13 Solutions