Question

What are the local extrema of `f(x) = 2 x + 3 /x`?

Answer 1

The local extrema are `-2sqrt(6)` at `x = -sqrt(3/2)`
and `2sqrt(6)` at `x = sqrt(3/2)`

Explanation:

Local extrema are located at points where the first derivative of a function evaluate to `0`. Thus, to find them, we will first find the derivative `f'(x)` and then solve for `f'(x) = 0`.

`f'(x) = d/dx(2x+3/x) = (d/dx2x) + d/dx(3/x) = 2 - 3/x^2`

Next, solving for `f'(x) = 0`

`2-3/x^2 = 0`

`=> x^2 = 3/2`

`=> x = +-sqrt(3/2)`

Thus, evaluating the original function at those points, we get

`-2sqrt(6)` as a local maximum at `x = -sqrt(3/2)`
and
`2sqrt(6)` as a local minimum at `x = sqrt(3/2)`