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?

Don't forget that critical points also occur where the derivative is undefined, as long as the point in question is in the domain of the original function.
Since $f(x) = x^{1/3} - (x+1)^{1/3}$, then $f'(x) = \dfrac13 x^{-2/3} - \dfrac13 (x+1)^{-2/3}$.
Note that $f'(x)$ is undefined at $x=0$ and $x=-1$, and both of these are in the domain of $f(x)$, so there may be extrema at those locations.
