On which intervals is f increasing?

Question

Let f be a function defined for all real numbers. 

Also, let f', the derivative of f, be defined as f'(x)=((x-2)^3))/x.

 

Options are 

(-Infinite, 0) and (2, infinite)

(-Infinite, 0) only

(-Infinite, 0) and (0,2)

(0,2) and (2, infinite)

All real numbers (excluding 0 and 2)

 

Answer 1

It is increasing when the derivative is greater than 0.  To determine when (x-2)^3/x > 0, we note that the numerator is 0 at x = 2, and the denominator is 0 when x = 0.  Next, we notice that the power in the numerator is 3, which is odd, and the power in the denominator is 1, which is odd.  This means that our interval is segmented at 0 and 2.

The function is positive at (2, inf), where the numerator and denominator are both negative, and at (-inf, 0), where the numerator and denominator are both positive.  It is negative on (0, 2).

The function increases on (-inf, 0) (2, inf), the first choice.