Finding Absolute Extrema Of Functions: Step-By-Step Guide With Example

Candidates Test (absolute extrema)

1. verify f is continuous on [a.b]2. find all critical points on [a,b]3. evaluate f at the critical points AND endpoints4. identify absolute extrema (largest y-value is the abs max, smallest y-value is the abs min)

The absolute extrema of a function are the highest and lowest points on the entire domain of the function. To determine the absolute extrema of a function, follow these steps:

1. Find the critical points of the function by setting its derivative equal to zero and solving for x.
2. Evaluate the function at each critical point as well as the endpoints of the domain.
3. Compare the values obtained in step 2 to determine which ones are the absolute maximum and minimum values.

Example: Find the absolute extrema of the function f(x) = x^3 – 3x^2 + 2 on the interval 0 ≤ x ≤ 2.

1. Find the critical points:
f'(x) = 3x^2 – 6x
3x(x-2) = 0
x = 0, 2 are the critical points.

2. Evaluate the function at each critical point and endpoints of the domain:
f(0) = 2
f(2) = 2
f(2) = 2
f(0) = 2

3. Compare the values:
The absolute maximum value of f(x) is 2, which occurs at both endpoints of the interval, while the absolute minimum value of f(x) is -2, which occurs at x = 1.

Therefore, the absolute extrema of f(x) on 0 ≤ x ≤ 2 are:
Absolute maximum value of 2 at x = 0 and x = 2
Absolute minimum value of -2 at x = 1

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