Chapter 4, Problem 85RE

### 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 In Exercises 81-92, analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. f ( x ) = 5 − 3 x x − 2

To determine

To graph: The provided function f(x)=53xx2.

Explanation

Given:

The function f(x)=5âˆ’3xxâˆ’2.

Graph:

Consider the provided function.

The function exists for all real values of x except 2. Thus the domain of the function is (âˆ’âˆž,2)âˆª(2,âˆž).

The range of the function would be (âˆ’âˆž,âˆ’3)âˆª(âˆ’3,âˆž).

Now find the x and y intercepts by equating f(x) and x to zero respectively to obtain:

The function has one x-intercepts (53,0) and the y-intercept is (0,âˆ’52).

The function has a horizontal asymptote y=âˆ’3 and a vertical asymptote x=2.

Now, differentiate the function with respect to x and equate it to zero to obtain the critical points.

1(xâˆ’2)2=0x=2

This is not a critical point but a point of discontinuity.

This gives two test intervals (âˆž,2),(2,âˆž).

Let 0âˆˆ(âˆ’âˆž,2).

f'(0)=1(0âˆ’2)2>0

The function is increasing in this interval.

Let 3âˆˆ(2,âˆž).

f'(3)=1(3âˆ’2)2>0

The function is increasing in this interval.

Now see the second derivative to see if there are any inflection points.

f''(x)=âˆ’2(xâˆ’2)3âˆ’2(xâˆ’2)3=0x=2

This is not an inflection point but a point of discontinuity

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