Note: when the derivative curve is equal to zero, the original function must be at a critical point, that is, the curve is changing from increasing to decreasing or visa versa.

What if the derivative is zero?

The derivative of a function, f(x) being zero at a point, p means that p is a stationary point. That is, not "moving" (rate of change is 0). There are a few things that could happen. Either the function has a local maximum, minimum, or saddle point.

Does the derivative of 0 exist?

At x=0 the derivative is undefined, so x^(1/3) is not differentiable, unless we exclude x=0. At x=0 the function is not defined so it makes no sense to ask if they are differentiable there. To be differentiable at a certain point, the function must first of all be defined there! ... So it is not differentiable there.