If a function f(x) is constant, it can be proven that its derivative f'(x) equals zero. Conversely, if f'(x) equals zero, it can be shown that f(x) must also be constant by applying the Mean Value Theorem. The theorem states that if f is continuous and differentiable on an interval, there exists a point where the derivative equals the average rate of change, which leads to the conclusion that f(a) equals f(b) for any two points a and b. Additionally, using differentials reinforces the idea that a zero derivative implies no change in the function's value. Thus, both directions of the argument confirm that a function with a zero derivative is indeed constant.