Is f(x)=x^{5/3}-5x^{2/3} defined over (-infty, 0]?

Thordiswl 2022-09-16 Answered
Is f ( x ) = x 5 / 3 5 x 2 / 3 defined over ( , 0 ]?
I've encountered this question: Find and describe all local extrema of f ( x ) = 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 ( , 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?
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Answers (1)

ruinsraidy4
Answered 2022-09-17 Author has 17 answers
Step 1
I suspect that x 2 / 3 = x 2 3 ..
This is an old story, since one usually defines x x α 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 exp ( x , α ) for the function x x α with domain ( 0 , + ) and a generic real exponent α.

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