A Step-by-Step Guide to Finding the Absolute Extrema of a Function with Example

Candidates Test (absolute extrem. a)

To find the absolute extrema of a function, follow these steps:

1

To find the absolute extrema of a function, follow these steps:

1. Determine the domain of the function. It is always important to know the range of values over which the function is defined.

2. Find the critical points by setting the derivative of the function equal to zero and finding any values where the derivative is undefined. These points could potentially be the locations of the maximum and minimum values.

3. Evaluate the function at the critical points and the endpoints of the given domain.

4. Compare the values obtained in step 3. The highest value represents the absolute maximum, and the lowest value represents the absolute minimum.

Let’s go through an example to illustrate this process:

Example:
Consider the function f(x) = x^3 – 4x^2 + 3x + 2 on the domain [-2, 3].

Step 1:
The domain is given as [-2, 3], so the function is defined for all x values between -2 and 3 (including the endpoints).

Step 2:
To find the critical points, we need to find where the derivative is equal to zero or undefined.
Taking the derivative of f(x), we get f'(x) = 3x^2 – 8x + 3.

Setting f'(x) = 0, we have 3x^2 – 8x + 3 = 0.
This quadratic equation can be factored as (3x – 1)(x – 3) = 0.
So, we have two critical points: x = 1/3 and x = 3.

Step 3:
Now, we need to evaluate the function at all critical points and endpoints.
Evaluate f(-2) = (-2)^3 – 4(-2)^2 + 3(-2) + 2 = 16 + 16 – 6 + 2 = 28.
Evaluate f(3) = (3)^3 – 4(3)^2 + 3(3) + 2 = 27 – 36 + 9 + 2 = 2.

Evaluate f(1/3) = (1/3)^3 – 4(1/3)^2 + 3(1/3) + 2 = 1/27 – 4/9 + 1 + 2 = 2 2/9.

Step 4:
Comparing the values:
f(-2) = 28 (endpoint), f(1/3) = 2 2/9 (critical point), f(3) = 2 (endpoint).

Therefore, the absolute maximum value is 28, which occurs at x = -2, and the absolute minimum value is 2, which occurs at x = 3.

So, the function has an absolute maximum of 28 at x = -2, and an absolute minimum of 2 at x = 3.

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