Finding Absolute Maximum and Minimum Values of a Function | Step-by-Step Guide

absolute max and min

In mathematics, the concepts of absolute maximum and minimum are used to find the largest and smallest values of a function over a specified interval or domain

In mathematics, the concepts of absolute maximum and minimum are used to find the largest and smallest values of a function over a specified interval or domain.

Let’s consider a function f(x) defined over a closed interval [a, b]. The absolute maximum of f(x) on this interval is the largest value that the function attains, while the absolute minimum is the smallest value.

To find the absolute maximum and minimum, we first need to determine the critical points of the function. These are the points where the derivative of the function is either zero or undefined. We also need to evaluate the function at the endpoints of the interval.

1. Find critical points:
To find critical points, we solve the equation f'(x) = 0 or f'(x) does not exist. These points may include local maximums and minimums, as well as points where the function changes concavity.

2. Evaluate function at endpoints:
Evaluate the function at the endpoints of the interval, f(a) and f(b).

3. Compare values:
Compare the values of the function at the endpoints and critical points to determine the absolute maximum and minimum.

– The largest value obtained from these evaluations is the absolute maximum.
– The smallest value obtained from these evaluations is the absolute minimum.

It is important to remember that the absolute maximum and minimum may not always exist. The function should be continuous over the interval [a, b] in order to guarantee the existence of absolute maximum and minimum values.

It is also worth noting that absolute maximum and minimum are global values, while local maximum and minimum refer to values within a specific interval.

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