Find all the critical numbers of y = x (lnx)^2 on the interval (0, infinity)

Obviously, two of the critical numbers are x = e^(-2) and x = 1

I also included zero and infinity as two other critical numbers. But my book didn't include them! I understood that infinity isn't considered as a critical number (new info for me) but what about zero??? lnx at zero, not to mention the endpoint of the interval, is undefined and doesn't that make the slope undefined as well?