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

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 4YT

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To find: All the relative extrema of f(x)=x2ex .

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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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All the relative extrema of f ( x ) = x 2 e x .

Solution Summary: The author explains how to find the relative extrema of f(x)=2ex.

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To find: All the relative extrema of f(x)=x2ex .

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