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?


  • $\begingroup$ Which function did you plot ? There is a minimum for $x=2$ not a maximum. $\endgroup$ – Claude Leibovici Oct 7 '15 at 10:03
  • $\begingroup$ 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. $\endgroup$ – Nizar Oct 7 '15 at 10:04
  • $\begingroup$ Why is it only defined on (0, +inf) ? $\endgroup$ – O-Marwan kenobi Oct 7 '15 at 10:29
  • $\begingroup$ Shouldn't the function be defined for all real numbers? $\endgroup$ – O-Marwan kenobi Oct 7 '15 at 11:26
  • $\begingroup$ For negative $x$ $x^{1/3}$ is multivalued. $\endgroup$ – Urgje Oct 7 '15 at 11:45

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