Critical Points of a Function
Points where a function f(x) is not differentiable and points where its derivative (differentiable coefficient) is zero are called the critical points of the function f(x).
Maximum and minimum values of a function f(x) can occur only at critical points. This however does not mean that the function will have maximum or minimum values at all critical points. Thus the points where maximum or minimum value occur are necessarily critical points but a function may or may not have maximum or minimum value at a critical point.