Chapter 4.2, Problem 2E

(a)

To determine

To find: The theorem guarantees the existence of an absolute maximum value and an absolute minimum value for f if f is a continuous function on $[a,b]$.

Answer to Problem 2E

The, Extreme value theorem guarantees that the existence of an absolute maximum value and absolute minimum value for f.

Explanation of Solution

Reason:

Extreme value theorem says that, “If f is a continuous function on a closed interval $[a,b]$, then f has a maximum and minimum over that interval”.

(b)

To determine

To explain: The steps to find the maximum and the minimum values of the continuous function.

Explanation of Solution

The steps to find the values of maximum and minimum values of the continuous function are as follows.

Step 1:

Find the first derivative $f^{\prime }\left(x\right)$ of the given function $f\left(x\right)$.

Step 2:

Take $f^{\prime }\left(x\right)=0$ and obtain the values (solutions) of x. The solutions are called critical numbers.

Step 3:

Check whether the critical number(s) obtained in Step 2 are lies in the given interval or not.

Step 4:

Consider the critical numbers that are lies in the given interval.

Step 5:

Substitute the critical number(s) considered in Step 4 in $f\left(x\right)$.and obtain the functional value. Also obtain the functional values at the end points of the given interval.

Step 6:

From the values obtained in Step 5, identify the value where the function attains its maximum, is called an absolute maximum of $f\left(x\right)$. And the value where the function attains its minimum is called an absolute minimum of $f\left(x\right)$.