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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Is   f  (  x  )  =      x          5              /            3        −  5      x          2              /            3       defined over   (  −  ∞  ,  0  ]?I&#039;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&#039;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?

ruinsraidy4 · Accepted Answer

Step 1I suspect that       x          2              /            3        =            x      2        3    ..This is an old story, since one usually defines   x  ↦      x    α   only for   x  &amp;gt;  0. However, notation is never given once and for all, so that we should be careful when we write mathematics.Step 2A 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.

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

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