Explanation Given: The function is f ( x ) = 4 x + 1 x . Definition used: (1) If the function gets larger and larger in magnitude without bound as x approaches the number k, then the line x = k is the vertical asymptote. (2) A function f ( x ) said to be symmetric about the origin if f ( − x ) = − f ( x ) . Calculation: The given function is f ( x ) = 4 x + 1 x . Domain: The given function f ( x ) = 4 x + 1 x is defined at all the real number except the point x = 0 . Thus, the domain of the function f ( x ) = 4 x + 1 x is ( − ∞ , 0 ) ∪ ( 0 , ∞ ) . Symmetry: The given function is f ( x ) = 4 x + 1 x . Substitute x = − x in f ( x ) , f ( − x ) = 4 × ( − x ) + 1 ( − x ) = − 4 x − 1 x = − ( 4 x + 1 x ) Thus, f ( − x ) = − f ( x ) . Therefore, from the definition stated above, the polynomial is symmetric about the origin. Intercept: The function f ( x ) intercepts y -axis at x = 0 . But the function f ( x ) = 4 x + 1 x is not defined at x = 0 . That is the function f ( x ) does not intercept y -axis at any point. The function f ( x ) = 4 x + 1 x is never zero. That is the function f ( x ) does not intercept x -axis at any point. Asymptote: Observe that the graph of the function f ( x ) = 4 x + 1 x tends to the line x = 0 as the value of x tends to 0. Thus x = 0 is the vertical asymptote for the function f ( x ) = 4 x + 1 x . Observe the limit lim x → ∞ 1 x = 0 . Thus, as the value of x tends to infinity, the the graph of the function f ( x ) = 4 x + 1 x tends to the line y = 4 x . Thus y = 4 x is the oblique asymptote for the function f ( x ) = 4 x + 1 x . Critical point: The given function is f ( x ) = 4 x + 1 x . Differentiate f ( x ) with respect to x , f ′ ( x ) = 4 − 1 x 2 To obtain critical points, f ′ ( x ) = 0 , 4 − 1 x 2 = 0 1 4 = x 2 x 2 − ( 1 2 ) 2 = 0 ( x − 1 2 ) ( x + 1 2 ) = 0 Thus, x = − 1 2 , and x = 1 2 . Thus, the critical points are at x = − 1 2 , and x = 1 2 . Substitute x = − 1 2 , and x = 1 2 in f ( x ) = 4 x + 1 x and obtain the critical points as ( − 1 2 , − 4 ) , and ( 1 2 , 4 ) . Relative Maxima and Relative Minima : The critical points are at x = − 1 2 , and x = 1 2

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 13.4, Problem 2YT

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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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Solution Summary: The function f(x)=4x+1x is symmetric about the origin.

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