Chapter 10.5, Problem 22E

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11th Edition

Ronald J. Harshbarger + 1 other

ISBN: 9781305108042

Textbook Problem

In Problems 19-24, a function and its first and second derivatives are given. Use these to find any horizontal and vertical asymptotes, critical points, relative maxima, relative minima, and points of inflection. Then sketch the graph of each function.

To determine

**To calculate:** The horizontal and vertical asymptote, relative minimum, relative maximum and points of inflection for the provided equation

Explanation

**Given Information:**

The provided equation is

**Formula used:**

A vertical asymptote of a function

A vertical asymptote of a rational function

A horizontal asymptote of a function

A horizontal asymptote of a rational function

1. A line

2. The line

3. Does not exist if the degree of the numerator is greater than the degree of the denominator.

To find relative maxima and minima of a equation,

1. Set the first derivative of the equation to zero,

2. Substitute the critical values into

3. Evaluate

(a) If

(b) If

(c) If

**Calculation:**

Consider the provided function,

Recall that a vertical asymptote of a rational function

Set the denominator of the function equal to zero.

Thus, vertical asymptote is

Consider

Thus, there is no horizontal asymptote.

Now, consider the first derivative of the function

Now, to obtain the critical values, set

Thus,

Thus, the critical points are

Substitute

Thus, the critical point is

Substitute

Thus, the critical point is

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