Question

What are the critical values, if any, of `f(x)= x^(3/4) - 2x^(1/4)`?

Answer 1

By the definition I am accustomed to, they are `0` and `4/9`.

Explanation:

A critical value of `f` is a value in the domain of `f` at which `f'` does not exists or `f'(x)` is `0`.

The domain of the function `f(x)=x^(3/4)-2x^(1/4)` is `[0,oo)`.

The derivative is `f'(x) =3/4x^(-1/4)-1/2x^(-3/4) = (3x^(1/2)-2)/(4x^(3/2))`

`f'` fails to exist at `x=0` and `f'(x) = 0` at `x=4/9`.

Both `0` and `4/9` are in the domain, so both are critical values.