Question

How do you find the intervals of increasing and decreasing using the first derivative given `y=2x+1/x`?

Answer 1

`(-oo, -1/sqrt(2))`
`(-1/sqrt(2), 1/sqrt(2))`
`(1/sqrt(2), oo)`

Explanation:

The function can be written as

`y= 2x + x^-1`

Which differentiates to:

`y' = 2 - x^-2 = 2 - 1/x^2`

A function will change from increasing to decreasing and vice versa when the derivative equals `0`.

`0 = 2 - 1/x^2`

`1/x^2 = 2`

`1/2 = x^2`

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

Now select a test point, let it be #x = 1#. Evaluating within the derivative gives a positive value which means the function is increasing at that Point. It also means that the function is decreasing on `-1/sqrt(2) < x < 1/sqrt(2)`.

Hopefully this helps!