Analyzing a function with cube roots I am analyzing this function:
\begin{align}
f(x) = \sqrt[3]{ x } - \sqrt[3]{x+1}
\end{align}
I proved it has a horizontal asymptote at y=0. However, on the graph, it has a minimum for x=-0.5 but I can't find that. First derivative has no zeroes, so the function has no stacionary points. Second derivative also has no zeroes. How do I find that minimum?

You must've taken the first derivative wrong:
$$f'(x)=\frac13(x^{-2/3}-(x+1)^{-2/3})$$
Setting $f'(x)=0$,
$$0=x^{-2/3}-(x+1)^{-2/3}$$
$$x^{-2/3}=(x+1)^{-2/3}$$
$$\pm x=x+1$$
$$x=-1/2$$
