Question

What are the critical values, of `f(x) = x^(1/3)*(8-x) in [0,8]`?

Answer 1

They are `0` and `2`.

Explanation:

`f(x) = x^(1/3)*(8-x) = 8x^(1/3)-x^(4/3)` `" "` for `0 <= x <= 8`

`f'(x) = 8/3 x^(-2/3)-4/3x^(1/3) = 8/(3root(3)(x^2))-(4root(3)x)/3`

`= (8-4x)/(3root(3)(x^2))`

`f'(x)` does not exist at `x=0` which is in the domain of `f`, so `0` is a critical number for `f`.

`f'(x) = 0` at `x=2` which is in the domain of `f`, so `2` is a critical number for `f`.