Question

How do you find the critical numbers for `f(x) = x^(1/3) (x+4)` to determine the maximum and minimum?

Answer 1

See below.

Explanation:

Rewrite: `f(x) = x^(4/3)+4x^(1/3)`

The domain of `f` is all real numbers.

Differentiate:

`f'(x) = 4/3 x^(1/3)+4/3 x^(-2/3)`

`= (4(x+1))/(3 root(3)x^2)`

`f'(x)` does not exists at `0` and `f'(x) = 0` at `-1`. Both are in the domain of `f`, so both are critical numbers.