Chapter 4.6, Problem 53E

### 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 Transcendental Function In Exercises 43-54, 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. g ( x ) = log 4 ( x − x 2 )

To determine

To graph: The function g(x)=log4(xx2)

Explanation

Given: The function g(x)=log4(xâˆ’x2).

Graph: The provided function can be rewritten as:

g(x)=ln(xâˆ’x2)ln4

The function being will be defined for only when the logarithmic function will have positive argument. Thus the domain of the function will be (0,1)

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

There are no y-intercepts or x-intercepts.

Also, the lines x=0 and xâˆ’1 are vertical asymptotes for the provided function as the logarithmic function is not defined at these values.

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

2xâˆ’1xln4xâˆ’1=02xâˆ’1=0x=0.5

This gives two test intervals (0,0.5),(0.5,1).

Let 0.25âˆˆ(0,0.5).

g'(0.25)=2(0.25)âˆ’10.25ln4(0.25)âˆ’1=âˆ’0.5âˆ’1>0

The function is increasing on this interval.

Let 0.75âˆˆ(0.5,1).

g'(0.75)=2(0.75)âˆ’10.75ln4(0.75)âˆ’1=0.50.75ln3âˆ’1<0

The function is decreasing on this interval.

This implies that the function has a maximum at maxima at 0.5.

Now find the value of the function at the critical point.

g(0.5)=ln(0

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