How do you find the absolute extreme value?

Finding the absolute extreme values (absolute maximum and absolute minimum) of a function on a closed interval involves a systematic process. This process guarantees that you'll identify the highest and lowest points of the function within that specific interval. Here's how:

Steps to Find Absolute Extreme Values

Based on the provided reference, here's a breakdown of the method:

1. Verify Continuity: Ensure that the function, f(x), is continuous on the closed interval [a, b]. Continuity is crucial because the Extreme Value Theorem guarantees the existence of absolute extrema on closed intervals for continuous functions.

2. Find Critical Points: Locate all critical points of f(x) within the interval [a, b]. Critical points are the points where the derivative of the function, f'(x), is either equal to zero or undefined. These points are potential locations for local maxima, local minima, or saddle points, and therefore, potential locations for absolute extrema.

3. Evaluate the Function: Evaluate the original function, f(x), at all the critical points found in step 2 that lie within the interval [a, b], and also evaluate f(x) at the endpoints of the interval, a and b.

4. Identify Absolute Extrema: Compare all the function values obtained in step 3.

   • The largest value is the absolute maximum of f(x) on the interval [a, b].
   • The smallest value is the absolute minimum of f(x) on the interval [a, b].

Practical Insights and Examples

Let's illustrate with a simple example:

Suppose we want to find the absolute extreme values of the function f(x) = x² - 4x + 5 on the interval [0, 3].

1. Continuity: f(x) is a polynomial, and polynomials are continuous everywhere. So, it's continuous on [0, 3].

2. Critical Points: Find the derivative: f'(x) = 2x - 4. Set f'(x) = 0: 2x - 4 = 0 => x = 2. So, x = 2 is the only critical point, and it lies within the interval [0, 3].

3. Evaluate:

   • f(0) = (0)² - 4(0) + 5 = 5
   • f(2) = (2)² - 4(2) + 5 = 1
   • f(3) = (3)² - 4(3) + 5 = 2

4. Identify: Comparing the values 5, 1, and 2, we find:

   • The absolute maximum is 5, which occurs at x = 0.
   • The absolute minimum is 1, which occurs at x = 2.

Importance of Closed Interval

It's important to note that this method specifically applies to functions on closed intervals. If the interval is open (e.g., (a, b)) or infinite, the function might not have absolute extrema, or finding them may require additional techniques involving limits.

Summary

Step	Description
1. Verify Continuity	Ensure f(x) is continuous on [a, b].
2. Find Critical Points	Find where f'(x) = 0 or is undefined within [a, b].
3. Evaluate Function	Calculate f(x) at critical points and endpoints a and b.
4. Identify Extrema	Determine the largest (absolute max) and smallest (absolute min) values.