Step-by-Step Guide to Find the Absolute Minimum of a Function over its Entire Domain

absolute (or global) minimum

The absolute (or global) minimum refers to the lowest value that a function can attain over its entire domain

The absolute (or global) minimum refers to the lowest value that a function can attain over its entire domain. In other words, it is the lowest point on the graph of the function.

To determine the absolute minimum of a function, we need to investigate the values of the function at critical points and endpoints of its domain. A critical point is where the derivative of the function is either zero or undefined.

Here’s a step-by-step process to find the absolute minimum:

1. Find the critical points of the function by setting its derivative equal to zero and solving for the x-values. Additionally, check if there are any points where the derivative is undefined.

2. Evaluate the function at all critical points and endpoints of the domain.

3. Compare the values obtained in step 2 to identify the lowest point. This will be the absolute minimum of the function.

It’s important to note that the absolute minimum may occur at either a critical point or an endpoint, or potentially at both.

For example, let’s consider the function f(x) = x^2 – 4x + 5 over the domain [-2, 4]:

1. Take the derivative of f(x) to find critical points:
f'(x) = 2x – 4
Setting f'(x) equal to zero: 2x – 4 = 0
Solving for x yields x = 2, which is a critical point.

2. Evaluate f(x) at critical points and endpoints:
f(-2) = (-2)^2 – 4(-2) + 5 = 17
f(4) = (4)^2 – 4(4) + 5 = 5
f(2) = (2)^2 – 4(2) + 5 = 1

3. Compare the values obtained in step 2:
The lowest value is f(2) = 1, which occurs at the critical point x = 2. Therefore, the absolute minimum for the function f(x) = x^2 – 4x + 5 over the domain [-2, 4] is 1.

By applying this process, one can determine the absolute minimum of a function on its entire domain.

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