Question

How do you find the critical points for `f(x) = x - 3ln(x)` and the local max and min?

Answer 1

`(3,3-3ln3)`

Explanation:

`f(x) = x - 3lnx`

Differentiating wrt `x`:
`f'(x)=1-3/x`

Differentiating again wrt `x`:
`f''(x) = -3/x^2(-1) = 3/x^2, (>0 AA x in RR)`

At critical points, `f'(x) = 0 => 1-3/x = 0`
`:. 3/x = 1`
`x = 3`

`f(3) = 3-3ln3`
`f''(3) >0`

Hence, there is one critical point `(3,3-3ln3)` which is a minimum.

graph{x-3lnx [-10, 10, -5, 5]}