Explanation Given: The function is f ( x ) = − 2 x 3 − 9 x 2 + 108 x − 10 . Definition used: A function f ( x ) said to be symmetric about the y -axis if f ( − x ) = f ( x ) Calculation: The given function is f ( x ) = − 2 x 3 − 9 x 2 + 108 x − 10 . Domain: The given function f ( x ) = − 2 x 3 − 9 x 2 + 108 x − 10 is defined for all real values. Thus, the domain of the function f ( x ) = − 2 x 3 − 9 x 2 + 108 x − 10 is ( − ∞ , ∞ ) . Symmetry: The given function is f ( x ) = − 2 x 3 − 9 x 2 + 108 x − 10 . Substitute x = − x in f ( x ) , f ( − x ) = − 2 ( − x ) 3 − 9 ( − x ) 2 + 108 ( − x ) − 10 = 2 x 3 − 9 x 2 − 108 x − 10 Thus, f ( − x ) ≠ f ( x ) . Therefore, use 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 ) = − 2 × 0 3 − 9 × 0 2 + 108 × 0 − 10 = − 10 That is the function f ( x ) intercepts y -axis at the point ( 0 , − 10 ) . Asymptote: The function f ( x ) = − 2 x 3 − 9 x 2 + 108 x − 10 has no asymptote. Critical point: The given function is f ( x ) = − 2 x 3 − 9 x 2 + 108 x − 10 . Differentiate f ( x ) with respect to x , f ′ ( x ) = − 6 x 2 − 18 x + 108 = − 6 ( x 2 + 3 x − 18 ) = − 6 ( x + 6 ) ( x − 3 ) To obtain critical points, f ′ ( x ) = 0 , − 6 ( x + 6 ) ( x − 3 ) = 0 ( x + 6 ) ( x − 3 ) = 0 Thus, x = 3 and x = − 6 . Thus, the critical points are at x = 3 and x = − 6 . Substitute x = 3 and x = − 6 in f ( x ) = − 2 x 3 − 9 x 2 + 108 x − 10 and obtain the critical points as ( − 6 , − 550 ) and ( 3 , 179 ) . Relative Maxima and Relative Minima : The critical points are at x = 3 and x = − 6 . Substitute x = 1 in f ′ ( x ) = − 6 ( x + 6 ) ( x − 3 ) , f ′ ( 1 ) = − 6 ( 1 + 6 ) ( 1 − 3 ) = − 6 × ( − 2 ) = 12 > 0 Thus, the function f ( x ) is increasing on the interval ( − 6 , 3 )

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

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