Question

How do you use the Extreme Value Theorem to determine the global extrema of the function `f(x)=5+ (x^2)/(x+2)` on the closed interval [-1,3]?

Answer 1

Use what some authors call the Closed Interval Method.

Explanation:

The Extreme Value Theorem does not really tell us how to find extrema, it only guarantees that for a function that is continuous on a closed interval, there are extrema.

Nevertheless, we can find the extrema.

They must occur at either a critical number (in the interval) or at an endpoint of the interval.

So the method is: find critical number in the interval, then evaluate `f` at these critical numbers and at the endpoints.

For `f(x)=5+ (x^2)/(x+2)` on `[-1,3]`, we get

`f'(x) = (x^2+4x)/(x+2)^2` So the critical numbers are `-4, 0`.

The only critical number in the interval is `0`

Evaluate:

`f(-1) = 6`
`f(0) = 5`
`f(3) = 6 4/5 = 6.8`

The maximum is `6 4/5` (at `x=3`)
The minimum is `5` (at `x=0`)