$f(x)=x^\frac{5}{3}-5x^\frac{2}{3}$
is the same as :
$f(x)=(\sqrt[3]x)^5-(\sqrt[3]{5x})^2$
Except, with the first equation, my calculator returns an error for negative values of $x$ (We are assuming $x\in\mathbb{R}$ here)
The second equation seems to have no problems with negative values of $x$.
Various graphing technologies have failed to provide a conclusive result.
Does it look like this?
or this?
or this?
It's driving me mad. I'm sure there's a simple explanation, or something obvious that I'm missing here.
EDIT: I think I made a mistake. $(\sqrt[3]{5x})^2$ should be $5(\sqrt[3]{x})^2$. That seems to clear up some of the issues.