Explanation Given: The function is f ( x ) = x 5 − 15 x 3 . Definition used: A function f ( x ) said to be symmetric about the origin if f ( − x ) = − f ( x ) . Calculation: The given function is f ( x ) = x 5 − 15 x 3 . Domain: The given function f ( x ) = x 5 − 15 x 3 is defined for all real values. Thus, the domain of the function f ( x ) = x 5 − 15 x 3 is ( − ∞ , ∞ ) . Symmetry: The given function is f ( x ) = x 5 − 15 x 3 . Substitute x = − x in f ( x ) , f ( − x ) = ( − x ) 5 − 15 ( − x ) 3 = − x 5 + 15 x 3 = − ( x 5 − 15 x 3 ) Thus, f ( − x ) = − f ( x ) . Therefore, from the definition stated above and conclude that the polynomial is symmetric about the origin. Intercept: The function f ( x ) intercepts y -axis at x = 0 . Substitute the value x = 0 in f ( x ) , f ( 0 ) = ( 0 ) 5 − 15 × ( 0 ) 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 . Thus, x 5 − 15 x 3 = 0 x 3 ( x 2 − 15 ) = 0 x 3 ( x − 15 ) ( x + 15 ) = 0 Thus, x = 15 , x = − 15 , and x = 0 . That is the function f ( x ) intercepts x -axis at the points ( − 15 , 0 ) , ( 15 , 0 ) , and ( 0 , 0 ) . Asymptote: The function f ( x ) = x 5 − 15 x 3 has no asymptote. Critical point: The given function is f ( x ) = x 5 − 15 x 3 . Differentiate f ( x ) with respect to x , f ′ ( x ) = 5 x 4 − 45 x 2 To obtain critical points, f ′ ( x ) = 0 , 5 x 4 − 45 x 2 = 0 5 x 2 ( x 2 − 9 ) = 0 x 2 ( x − 3 ) ( x + 3 ) = 0 Thus, x = 3 , x = − 3 , and x = 0 . Thus, the critical points are at x = 3 , x = − 3 , and x = 0 . Substitute x = 3 , x = − 3 , and x = 0 in f ( x ) = x 5 − 15 x 3 and obtain the critical points as ( 3 , − 162 ) , ( − 3 , 162 ) , and ( 0 , 0 ) . Relative Maxima and Relative Minima : The critical points are at x = 3 , x = − 3 , and x = 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 10E

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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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Solution Summary: The function f(x) is symmetric about the origin if it is defined for all real values.

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