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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.

I've managed to find the extrema, but I am not sure whether the function is defined on $(-\infty, 0]$. To make sure I looked on the internet at some graphing calculators and some of them graphed the function on that interval while others did not. Is it defined on that interval?

rubik
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AlexandreAlvard
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1 Answers1

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I suspect that $$x^{2/3}=\sqrt[3]{x^2}.$$ This is an old story, since one usually defines $x \mapsto x^\alpha$ only for $x>0$. However, notation is never given once and for all, so that we should be careful when we write mathematics.

A possible solution would be to reserve something like $\exp(x,\alpha)$ for the function $x \mapsto x^\alpha$ with domain $(0,+\infty)$ and a generic real exponent $\alpha$. But nobody does this...

rubik
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Siminore
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  • Sorry, I don't really understand what you mean... Are you saying it is not defined? – AlexandreAlvard Oct 13 '15 at 08:41
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    I am saying that it is up to you. If you decide that $x^{2/3}=\sqrt[3]{x^2}$, then it is clearly defined. If you insist that $x^{2/3}=e^{\frac23 \log x}$, then it is undefined. – Siminore Oct 13 '15 at 08:44