Steps to Find Absolute Extrema of a Function: A Comprehensive Guide

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To find the absolute extrema of a function, we need to identify the highest and lowest points of the function within a given interval or on its entire domain

To find the absolute extrema of a function, we need to identify the highest and lowest points of the function within a given interval or on its entire domain. Here are the steps you can follow to solve a problem related to finding absolute extrema:

1. Identify the domain: Determine the range of values for which the function is defined. For example, if the function is defined only for x values between 0 and 10, then the domain would be [0, 10].

2. Find the critical points: Critical points are points on the graph where the derivative of the function is either zero or does not exist. To find critical points, calculate the derivative of the function and solve for x values where the derivative is zero or undefined.

3. Evaluate the function at critical points and the endpoints: Plug in the critical points as well as the endpoints of the given interval into the original function to find their corresponding function values.

4. Compare the function values: Compare the function values calculated at the critical points and endpoints to determine the highest (maximum) and lowest (minimum) function values within the given interval or domain.

Here’s an example problem to illustrate the steps:

Problem: Find the absolute extrema of the function f(x) = x^2 – 4x + 3 on the interval [0, 4].

Solution:
1. Domain: The given interval is [0, 4], so the function is defined for all x values from 0 to 4.

2. Critical points: Calculate the derivative of the function f(x) to find the critical points.
f'(x) = 2x – 4

To find critical points, set the derivative equal to zero:
2x – 4 = 0
2x = 4
x = 2

So, x = 2 is the critical point.

3. Evaluate the function: Plug in the critical point and the endpoints of the interval into the original function.
For x = 0: f(0) = (0)^2 – 4(0) + 3 = 3
For x = 2: f(2) = (2)^2 – 4(2) + 3 = -1
For x = 4: f(4) = (4)^2 – 4(4) + 3 = 3

4. Compare function values: Comparing the function values, we can conclude that the minimum value within the interval [0, 4] is -1 at x = 2, and the maximum value is 3 at both x = 0 and x = 4.

So, the absolute minimum of the function is -1 at x = 2, and the absolute maximum is 3 at both x = 0 and x = 4.

This method can be applied to any given function to find the absolute extrema within a specified interval or on the entire domain.

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