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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 4, Problem 2RCC

**(a)**

To determine

**To explain:** The Extreme value theorem.

Expert Solution

The extreme value theorem is stated as follow.

**Extreme Value Theorem:**

“If *f* is a continuous function on
*f* attains its absolute maximum value
*c* and *d* in the interval

From the following Figure 1, the meaning of extreme value theorem can be easily understood.

In the Figure 1 the function *f* is defined on a closed interval

**(b)**

To determine

**To explain:** The working of Closed interval method.

Expert Solution

Closed interval method is an algorithm that is used to find the absolute minimum and the absolute maximum of a function over a closed interval

It consists of following three steps.

**Step 1:** Find the values of function *f* at the critical points in

**Step 2:** Find the values of function *f* at the end points of the interval

**Step 3:** Largest of the values from step 1 and step 2 is the absolute maximum value and lowest one is absolute minimum value.

Apply the closed interval method to find the absolute maximum and minimum of a function as illustrated in following example.

**Example:**

The absolute maximum and absolute minimum of the function

**Calculation**:

Obtain the first derivative of the given function.

Set

Here, the critical number

**Step 1:** Find the values of function *f* at the critical points in

Here the critical point is

**Step 2:** Apply the extreme values of the given interval in

Substitute

Substitute

**Step 3:**

Since the largest functional value is the absolute maximum and the smallest functional value is the absolute minimum, the absolute maximum of

Therefore, the absolute maximum value of