Alright, I've been thinking of going back to school and have been reviewing some calculus I took years ago. I'm having a bit of a problem with a First Derivative Test question and hoping someone can help me out. As I understand the first derivative test I am to take the derivative of a given function f(x), set the result to zero, and find the value(s) of x that makes f'(x) = 0. these are the critical points where the graph of f(x) *might* change direction. so I have this seemingly harmless function to work on:

f(x) = x^{3}- (3/2)x^{2}+ 2x - 3

I can derive that pretty easily: f'(x) = 3x^{2}- 3x + 2

when I set this to zero I get 3x^{2}- 3x + 2 = 0 -> x^{2}- x + 2/3 = 0

Now my troubles begin, I can't factor that directly, if I use the quadratic formula I get a negative in the square root, that leaves me with completing the square. I can do that but am not sure what to do with that in order to find the critical points. When I go back to the intent of the problem I am really to find where f(x) increases and where it decreases. The critical points are supposed to tell me where directional changes occur. But I recognize that since this f(x) is a cubic it increases everywhere. If I did not know that I'd be lost. Can someone help me find my way?