To determine To calculate: The absolute extremum of given function g ( x ) = x e x on interval ( 0 , 3 ) . Explanation Given: The function is g ( x ) = x e x . Formula used: If f ( x ) = u ( x ) v ( x ) , if all indicated derivatives exist, and if v ( x ) ≠ 0 , then f ′ ( x ) = v ( x ) ⋅ u ′ ( x ) − u ( x ) ⋅ v ′ ( x ) [ v ( x ) ] 2 and Critical point theorem, if a function f ( x ) is continuous on a given interval ( a , b ) and has only one critical number in the interval ( a , b ) , say, x = c . 1. If f ( x ) has a local maximum at x = c , then it will be the absolute maxima of f ( x ) on the interval ( a , b ) . 2. If f ( x ) has a local minimum at x = c , then it will be the absolute minima of f ( x ) on the interval ( a , b ) . Calculation: The given function is, g ( x ) = x e x Differentiate the above function with respect to x . g ′ ( x ) = ( 1 ) e x − x e x e 2 x = e x − x e x e 2 x To find the critical numbers, put g ′ ( x ) = 0 . This implies, e x − x e x e 2 x = 0 Simplify the above expression. e x ( 1 − x e 2 x ) = 0 1 − x = 0 x = 1 Therefore, x = 1 The only critical number in the interval ( 0 , 3 ) is x = 1

12th Edition

ISBN: 8220106844557

Author: MULLINS

Publisher: PEARSON

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Chapter 12.3, Problem 25E

To determine

To calculate: The absolute extremum of given function g(x)=xex on interval (0,3).

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Chapter 12.3, Problem 24E

Chapter 12.3, Problem 26E

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