Question

How do you determine the absolute maximum and minimum values of a continuous function on a closed​ interval?

Answer 1

To find the absolute maximum and minimum values of a continuous function on a closed interval, you must calculate the derivative to find critical points, evaluate the function at those points and the endpoints, then compare these values.

Step-by-step explanation:

To determine the absolute maximum and minimum values of a continuous function on a closed interval, you need to follow these steps:

1. Calculate the derivative of the function to find its critical points.
2. Evaluate the function at each critical point within the interval.
3. Also, evaluate the function at the endpoints of the interval.
4. Compare all these values to determine which is the absolute maximum and which is the absolute minimum on the interval.

The absolute maximum is the highest value obtained, and the absolute minimum is the lowest value obtained from the function at these points.

Moreover, in the context of continuous probability density functions, these methods apply to determining maximum and minimum values of probabilities within a given range,

Answer 2

Steps are detailed below.

Step-by-step explanation:

The steps to determine the absolute maximum and minimum values of a continuous function on a closed​ interval are;

For example, if we have a function f(x) = x³ - 3x + 2 on the interval (-3, 4)

The steps are;

Step 1:

We will find the first derivative of f(x) which is; f'(x) = 3x² - 3.

Then we will find the critical values of this derivative. The critical values are the values of x that makes f'(x) = 0. But these critical values must fall in between the given interval (-3, 4)

Step 2:

Plug in the critical values gotten in step 1 above into the original f(x) function and note down the answers.

Step 3;

Plug in the endpoint values on the interval (-3, 4) given in the question into the original f(x) function and note down the answers.

Step 4;

The highest and lowest values of f(x) gotten in steps 2 & 3 are the absolute maximum and absolute minimum values respectively of the continuous function f(x).