Explanation Given: The function is f ( x ) = x 2 3 − x 5 3 . Definition used: A function f ( x ) is said to be symmetric about the y -axis if f ( − x ) = f ( x ) Calculation: The given function is f ( x ) = x 2 3 − x 5 3 . Domain: The given function f ( x ) = x 2 3 − x 5 3 is defined for all real values. Thus, the domain of the function f ( x ) = x 2 3 − x 5 3 is ( − ∞ , ∞ ) . Symmetry: The given function is f ( x ) = x 2 3 − x 5 3 . Substitute x = − x in f ( x ) , f ( − x ) = ( − x ) 2 3 − ( − x ) 5 3 = x 2 3 + x 5 3 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 ) 2 3 − ( 0 ) 5 3 = 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 2 3 − x 5 3 = 0 x 2 3 ( 1 − x ) = 0 Thus, x = 0 and x = 1 . That is the function f ( x ) intercepts x -axis at the point ( 0 , 0 ) and ( 1 , 0 ) . Asymptote: The function f ( x ) = x 2 3 − x 5 3 has no asymptote. Critical point: The given function is f ( x ) = x 2 3 − x 5 3 . Differentiate f ( x ) with respect to x , f ′ ( x ) = 2 3 x − 1 3 − 5 3 x 2 3 = 1 3 x − 1 3 ( 2 − 5 x ) = 2 − 5 x 3 x 1 3 To obtain critical points, f ′ ( x ) = 0 , 2 − 5 x 3 x 1 3 = 0 2 − 5 x = 0 x = 2 5 Thus, x = 2 5 . Thus, the critical point is at x = 2 5 . Substitute x = 2 5 in the function f ( x ) = x 2 3 − x 5 3 to obtain the critical point as ( 2 5 , 0.326 ) . Relative Maxima and Relative Minima : The critical point is at x = 2 5 . But f ′ ( x ) = 2 − 5 x 3 x 1 3 is not defined at the point x = 0 . Substitute x = 1 in f ′ ( x ) = 2 − 5 x 3 x 1 3 , f ′ ( 1 ) = 2 − 5 × 1 3 ( 1 ) 1 3 = − 3 3 = − 1 < 0 Thus, the function f ( x ) is decreasing on the interval ( 2 5 , ∞ ) . Substitute x = − 1 in f ′ ( x ) = 2 − 5 x 3 x 1 3 , f ′ ( − 1 ) = 2 − 5 × ( − 1 ) 3 ( − 1 ) 1 3 = 3 − 3 < 0 Thus, the function f ( x ) is decreasing on the interval ( − ∞ , 0 )

Finite Mathematics and Calculus with Applications

1st Edition

ISBN: 9781323188361

Author: Margaret Lial

Publisher: Pearson Education

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Chapter 13.4, Problem 29E

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

Solution Summary: The author explains that the function f(x) is symmetric about the y -axis, and concludes that it has no symmetry.

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