Chapter 4.6, Problem 55E

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Chapter
Section

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

# Analyzing the Graph of a Function Using TechnologyIn Exercises 55-62, use a computer algebra system to analyze and graph the function. Identify any relative extrema, points of inflection, and asymptotes. f ( x ) = 20 x x 2 + 1 − 1 x

To determine

To graph: The provided function f(x)=20xx2+11x with the help of a graphing utility and to obtain the extremas, inflection points and asymptotes for the function.

Explanation

Given:

f(x)=20xx2+1âˆ’1x

Graph:

The graph of the function can be obtained using Maple.

Maple Command:

Maple Graph:

Now consider the provided function.

f(x)=20xx2+1âˆ’1x

Cleary, the function is not defined at 0 due to the term 1x which implies that the function has a vertical asymptote as x=0.

Now obtain the limit of the function at infinity.

limxâ†’âˆžf(x)=limxâ†’âˆž20xx2+1âˆ’limxâ†’âˆž1x=limxâ†’âˆž20x1+1x2âˆ’0=01+0âˆ’0=0 dividing numerator and denominator by x2

This implies that the function has a horizontal asymptote as y=0.

Differentiate the function with respect to x and equate that to zero to get the critical points.

f'(x)=20(1âˆ’x2)(x2+1)2+1x220(1âˆ’x2)(x2+1)2+1x2=019x4+22x2+1=0xâ‰ˆÂ±1.10

Now check the value of the second derivative at the critical points.

f''(x)=92(19x6âˆ’63x4âˆ’3x2âˆ’1)x3(x2+1)3f''(âˆ’1.1)=92(19(âˆ’1.1)6âˆ’63(âˆ’1.1)4âˆ’3(âˆ’1.1)2âˆ’1)(âˆ’1.1)3((âˆ’1.1)2+1)3<0

This implies that the function has a relative maximum here.

And,

f''(x)=92(19x6âˆ’63x4âˆ’3x2âˆ’1)x3(x2+1)3f''(1

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