Question

How do you find the critical points for `f(x)= x^(4/3) + 4x^(1/3)+ 4x^(-2/3)`?

Answer 1

The critical points are `(1,f(1))=(1,9)`

Explanation:

First the domain of f(x) is `D(f)=(0,+oo)` hence

we must find the roots of the first derivative which is

`(df(x))/dx=(4*(x^2+x-2))/(3x^(5/3))`

The roots of the derivative are the roots of the equation `x^2+x-2=0` hence `x=-2` , `x=1`.

But `x=-2` does mot belong in the domain of f(x).