Question

What are the absolute extrema of `f(x)=x-sqrt(5x-2) in(2,5)`?

Answer 1

There are no absolute extrema in the interval `(2, 5)`

Explanation:

Given: `f(x) = x - sqrt(5x - 2) in (2, 5)`

To find absolute extrema we need to find the first derivative and perform the first derivative test to find any minimum or maximums and then find the `y` values of the end points and compare them.

Find the first derivative:

`f(x) = x - (5x - 2)^(1/2)`

`f'(x) = 1 - 1/2( 5x - 2)^(-1/2)(5)`

`f'(x) = 1 - 5/(2sqrt(5x - 2))`

Find critical value(s) `f'(x) = 0`:

`1 - 5/(2sqrt(5x - 2)) = 0`

`1 = 5/(2sqrt(5x - 2))`

`2sqrt(5x - 2) = 5`

`sqrt(5x - 2) = 5/2`

Square both sides: `5x - 2 = +- 25/4`

Since the domain of the function is limited by the radical:

`5x - 2 >= 0; " "x >= 2/5`

We only need to look at the positive answer:

`5x - 2 = + 25/4`

`5x = 2/1 *4/4 + 25/4 = 33/4`

`x = 33/4 * 1/5 = 33/20 ~~1.65`

Since this critical point is `< 2`, we can ignore it.

This means the absolute extrema are at the endpoints, but the endpoints are not included in the interval.