Explanation Given: The function is f ( x ) = x 4 − 4 x 3 . 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 − 4 x 3 . Domain: The given function f ( x ) = x 4 − 4 x 3 is defined for all real values. Thus, the domain of the function f ( x ) = x 4 − 4 x 3 is ( − ∞ , ∞ ) . Symmetry: The given function is f ( x ) = x 4 − 4 x 3 . Substitute x = − x in f ( x ) , f ( − x ) = ( − x ) 4 − 4 × ( − x ) 3 = x 4 + 4 x 3 Thus, f ( − x ) ≠ f ( x ) . Therefore, from the definition stated above and conclude that the polynomial has no symmetry 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 − 4 × ( 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 4 − 4 x 3 = 0 x 3 ( x − 4 ) = 0 Thus, x = 4 , and x = 0 . That is the function f ( x ) intercepts x -axis at the points ( 4 , 0 ) , and ( 0 , 0 ) . Asymptote: The function f ( x ) = x 4 − 4 x 3 has no asymptote. Critical point: The given function is f ( x ) = x 4 − 4 x 3 . Differentiate f ( x ) with respect to x , f ′ ( x ) = 4 x 3 − 12 x 2 = 4 x 2 ( x − 3 ) To obtain critical points, f ′ ( x ) = 0 , 4 x 2 ( x − 3 ) = 0 x 2 ( x − 3 ) = 0 Thus, x = 3 and x = 0 . Thus, the critical points are at x = 3 and x = 0 . Substitute x = 3 and x = 0 in f ( x ) = x 4 − 4 x 3 and obtain the critical points as ( 3 , − 27 ) and ( 0 , 0 ) . Relative Maxima and Relative Minima: The critical points are at x = 3 and x = 0 . Substitute x = 1 in f ′ ( x ) = 4 x 2 ( x − 3 ) , f ′ ( 1 ) = 4 × ( 1 ) 2 ( 1 − 3 ) = 4 × ( − 2 ) = − 8 < 0 Thus, the function f ( x ) is decreasing on the interval ( − ∞ , 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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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topic…

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Chapter 13.4, Problem 9E

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To sketch: The graph of the function.

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1. 2. In Exercises 1, 2, 3, 4, 5, 6, 7, and 8, use the graph of the function f to find lim f(x), lim f(x), and xa x+a+ lim f(x) at the indicated value of a, if the limit exists. x a Answ 2: the init X1st

1. 2. In Exercises 1, 2, 3, 4, 5, 6, 7, and 8, use the graph of the function f to find lim f(x), lim f (x), and xa x→a+ lim f (x) at the indicated value of a, if the limit exists. xa Ans 2: the lin X1st S 4+ 3 y = f(x) 2 1 2 3 4 5 6 a=3

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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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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Chapter 13 Solutions