Chapter 10.1, Problem 31E

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11th Edition

Ronald J. Harshbarger + 1 other

ISBN: 9781305108042

Textbook Problem

In Problems 31-36, both a function and its derivative are given. Use them to find critical values, critical points, intervals on which the function is increasing and decreasing, relative maxima, relative minima, and horizontal points of inflection; sketch the graph of each function.

To determine

**To calculate:** The critical values, critical points, the relative maxima, relative minima, or horizontal points of inflection of function

Explanation

**Given Information:**

The provided function is

**Formula Used:**

The simple power rule of derivative,

To obtain the relative maxima and minima of a function,

1. Find the first derivative of the function.

2. Set the derivative equal to

3. Evaluate

(a) If

(b) If

**Calculation:**

Consider the provided function

The critical values are the only values at which the graph can have turning points, the derivative cannot change sign anywhere except at the critical value.

Hence, there will no change in the values of critical values as in the derivative graph.

Now, equate the derivative to 0.

From the property of multiplication,

Or,

Or,

Therefore, the critical values are

Substitute 0 for *x* in the function

Therefore,

Substitute 1 for *x* in the function

Therefore,

Substitute 2 for *x* in the function

Therefore,

Test the derivative at the left and right sides of all the critical points.

Therefore the intervals to check the maxima, minima and point of inflection are

For interval

Substitute *x* in the function

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