# Find and describe all relative extrema

I've encountered this question: Find and describe all local extrema of $f(x) = x^{5 /3} − 5x^{2/3}$. Also indicate on which regions the function is increasing and decreasing.

Using the derivative, I found the minimum at x = 2. But when I graphed the function, I saw that it goes up to zero and seems to have a maximum here. Is it really an extrema? Why didn't it show up with the derivative?

Thanks!

• Which function did you plot ? There is a minimum for $x=2$ not a maximum. – Claude Leibovici Oct 7 '15 at 10:03
• In fact you function is defined on $[0, \infty)$, and its derivative is defined in $(0, \infty)$. So when you find the derivative you check for extrema in the interior of the domain of $f$, you still need to check if you have extrama at the boundaries. So you are right $0$ is a local max. – Nizar Oct 7 '15 at 10:04
• Why is it only defined on (0, +inf) ? – O-Marwan kenobi Oct 7 '15 at 10:29
• Shouldn't the function be defined for all real numbers? – O-Marwan kenobi Oct 7 '15 at 11:26
• For negative $x$ $x^{1/3}$ is multivalued. – Urgje Oct 7 '15 at 11:45