Explanation Given: The function is f ( x ) = − x 3 + 3 x 2 + 9 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 ) = − x 3 + 3 x 2 + 9 x − 10 . Domain: The given function f ( x ) = − x 3 + 3 x 2 + 9 x − 10 is defined for all real values. Thus, the domain of the function f ( x ) = − x 3 + 3 x 2 + 9 x − 10 is ( − ∞ , ∞ ) . Symmetry: The given function is f ( x ) = − x 3 + 3 x 2 + 9 x − 10 . Substitute x = − x in f ( x ) , f ( − x ) = − ( − x ) 3 + 3 × ( − x ) 2 + 9 ( − x ) − 10 = x 3 + 3 x 2 − 9 x − 10 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 ) 3 + 3 ( 0 ) 2 + 9 ( 0 ) − 10 = − 10 That is the function f ( x ) intercepts y -axis at the point ( 0 , − 10 ) . Asymptote: The function f ( x ) = − x 3 + 3 x 2 + 9 x − 10 has no asymptote. Critical point: The given function is f ( x ) = − x 3 + 3 x 2 + 9 x − 10 . Differentiate f ( x ) with respect to x , f ′ ( x ) = − 3 x 2 + 6 x + 9 To obtain critical points, f ′ ( x ) = 0 , − 3 x 2 + 6 x + 9 = 0 − 3 ( x 2 − 2 x − 3 ) = 0 ( x + 1 ) ( x − 3 ) = 0 Thus, x = 3 and x = − 1 . Thus, the critical points are at x = 3 and x = − 1 . Substitute x = 3 and x = − 1 in f ( x ) = − x 3 + 3 x 2 + 9 x − 10 and obtain the critical points as ( − 1 , − 15 ) and ( 3 , 17 ) . Relative Maxima and Relative Minima : The critical points are at x = 3 and x = − 1 . Substitute x = − 2 in f ′ ( x ) = − 3 x 2 + 6 x + 9 , f ′ ( − 2 ) = − 3 × ( − 2 ) 2 + 6 × ( − 2 ) + 9 = − 12 − 12 + 9 = − 15 < 0 Thus, the function f ( x ) is decreasing on the interval ( − ∞ , − 1 )

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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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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The graph of the function.

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

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