Explanation Given: The function is, f ( x ) = x e 2 x . Definition used: A function f ( x ) is said to be symmetric about the y -axis if f ( − x ) = f ( x ) . Calculation: Domain: The given function f ( x ) = x e 2 x is defined for all real values. Thus, the domain of the function f ( x ) = x e 2 x is ( − ∞ , ∞ ) . Symmetry: The given function is f ( x ) = x e 2 x . Substitute x = − x in f ( x ) , f ( − x ) = ( − x ) e 2 ( − x ) = − x e − 2 x Thus, f ( − x ) ≠ f ( x ) . Therefore, from the definition stated above and conclude that the polynomial has no symmetry. Intercept: The function f ( x ) intercepts y -axis at x = 0 . Substitute the value x = 0 in f ( x ) , f ( 0 ) = ( 0 ) e 2 ( 0 ) = 0 That is the function f ( x ) intercepts y -axis at the point ( 0 , 0 ) . The function f ( x ) intercepts x -axis at f ( x ) = 0 . Substitute the value f ( x ) = 0 in f ( x ) , x e 2 x = 0 The function e 2 x is never 0. Thus, x = 0 . That is the function f ( x ) intercepts x -axis at the point ( 0 , 0 ) . Asymptote: The given function is f ( x ) = x e 2 x . Calculate the limit of the function, lim x → ∞ f ( x ) = lim x → ∞ x e 2 x = 0 ( ∵ lim x → ∞ x n e n x = 0 ) Thus, the function f ( x ) = x e 2 x has horizontal asymptote at y = 0 . Critical point: The given function is f ( x ) = x e 2 x . Differentiate f ( x ) with respect to x , f ′ ( x ) = ( 1 ) ( e 2 x ) + ( x ) ( 2 e 2 x ) = e 2 x ( 2 x + 1 ) To obtain critical points, f ′ ( x ) = 0 , e 2 x ( 2 x + 1 ) = 0 2 x + 1 = 0 x = − 1 2 Thus, x = − 1 2 . Thus, the critical point is at x = − 1 2 . Substitute x = − 1 2 in the function f ( x ) = x e 2 x to obtain the critical point as ( − 1 2 , − 1 2 e )

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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Differential Equations

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Chapter 13, Problem 53RE

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To sketch: The graph of the function f(x)=xe2x .

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Chapter 13, Problem 52RE

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Finite Mathematics and Calculus with Applications (10th Edition)

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The graph of the function f ( x ) = x e 2 x .

Solution Summary: The author explains the function f(x)=xe2x and concludes that the polynomial has no symmetry.

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To sketch: The graph of the function f(x)=xe2x .

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Chapter 13 Solutions