MHB Getting a sign chart for a function

[tmt1]

I have the function

$$\frac{5(1-x)}{3x^{1/3}}$$

for which I need to find a sign chart. I know that for $x = 0$ and $x = 1$ are the inflection points, since those are the points for which the numerator and denominator will equal zero.

So, is the function positive or negative when $x < 0$, $x > 1$, and $0 < x < 1$?. I can get the values for when $x > 0$ easily enough, but what about when $x < 0$?

If I take $-1$, then
$$\frac{10}{3(-1)^{1/3}}$$

But I though for roots the radicand can't be negative?

 

[MarkFL]

Odd roots (not to be confused with the zeroes of a function) can be negative, since the product of an odd number of negative numbers is negative. :)

 

[HallsofIvy]

The numerator, 5(1- x), is positive for x< 1 and negative for x> 1. The denominator, 3x^{1/3}, is negative for x< 0 and positive for x> 0. A fraction is positive as long as both numerator and denominator have the same sigh, negative if they have different signs.

For x< 0< 1, the numerator is positive and the denominator is negative.

For 0< x< 1, the numerator is still positive and the denominator is positive.

For 0< 1< x, the numerator is negative and the denominator is positive.