Explanation Given: The function is f ( x ) = x 4 − 24 x 2 + 80 . 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 4 − 24 x 2 + 80 . Domain: The given function f ( x ) = x 4 − 24 x 2 + 80 is defined for all real values. Thus, the domain of the function f ( x ) = x 4 − 24 x 2 + 80 is ( − ∞ , ∞ ) . Symmetry: The given function is f ( x ) = x 4 − 24 x 2 + 80 . Substitute x = − x in f ( x ) , f ( − x ) = ( − x ) 4 − 24 × ( − x ) 2 + 80 = x 4 − 24 x 2 + 80 = f ( x ) Thus, f ( − x ) = f ( x ) . Therefore, from the definition stated above and conclude that the polynomial is symmetric about y -axis. Intercept: The function f ( x ) intercepts y -axis at x = 0 . Substitute the value x = 0 in f ( x ) , f ( 0 ) = ( 0 ) 4 − 24 × ( 0 ) 2 + 80 = 80 That is the function f ( x ) intercepts y -axis at the point ( 0 , 80 ) . The function f ( x ) intercepts x -axis at f ( x ) = 0 . Thus, x 4 − 24 x 2 + 80 = 0 ( x 2 − 4 ) ( x 2 − 20 ) = 0 ( x − 2 ) ( x + 2 ) ( x 2 − ( 2 5 ) 2 ) = 0 ( x − 2 ) ( x + 2 ) ( x − 2 5 ) ( x + 2 5 ) = 0 Thus, x = 2 , x = − 2 , x = 2 5 , and x = − 2 5 . That is the function f ( x ) intercepts x -axis at the points ( 2 , 0 ) , ( − 2 , 0 ) , ( 2 5 , 0 ) , and ( − 2 5 , 0 ) . Asymptote: The function f ( x ) = x 4 − 24 x 2 + 80 has no asymptote. Critical point: The given function is f ( x ) = x 4 − 24 x 2 + 80 . Differentiate f ( x ) with respect to x , f ′ ( x ) = 4 x 3 − 48 x = 4 x ( x 2 − 48 ) To obtain critical points, f ′ ( x ) = 0 , 4 x ( x 2 − 48 ) = 0 4 x ( x − 48 ) ( x + 48 ) = 0 x ( x − 2 3 ) ( x + 2 3 ) = 0 Thus, x = 2 3 , x = − 2 3 and x = 0 . Thus, the critical points are at x = 2 3 , x = − 2 3 and x = 0 . Substitute x = 2 3 , x = − 2 3 and x = 0 in f ( x ) = x 4 − 24 x 2 + 80 and obtain the critical points as ( 2 3 , − 64 ) , ( − 2 3 , − 64 ) and ( 0 , 80 )

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ISBN: 8220102020252

Author: RITCHEY

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Solution Summary: The function f(x) is said to be symmetric about the y -axis.

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