Thordiswl

2022-09-16

Is $f\left(x\right)={x}^{5/3}-5{x}^{2/3}$ defined over $\left(-\mathrm{\infty },0\right]$?
I've encountered this question: Find and describe all local extrema of $f\left(x\right)={x}^{5/3}-5{x}^{2/3}.$.
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 $\left(-\mathrm{\infty },0\right]$. 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?

ruinsraidy4

Step 1
I suspect that ${x}^{2/3}=\sqrt[3]{{x}^{2}}.$.
This is an old story, since one usually defines $x↦{x}^{\alpha }$ only for $x>0$. However, notation is never given once and for all, so that we should be careful when we write mathematics.
Step 2
A possible solution would be to reserve something like $\mathrm{exp}\left(x,\alpha \right)$ for the function $x↦{x}^{\alpha }$ with domain $\left(0,+\mathrm{\infty }\right)$ and a generic real exponent $\alpha$.

Do you have a similar question?