But you haven't proved whether the function has the maximum or the minimum at $x=e$.

For this, you have to take $2^{\mathrm{nd}}$ derivative as well. By the way, the function has the maximum at $x=e$, because $f''(e)<0$.

For this, you have to take $2^{\mathrm{nd}}$ derivative as well. By the way, the function has the maximum at $x=e$, because $f''(e)<0$.