It follows from these definitions that the function f(x) has a critical point x0 with critical value y0, if and only if (x0, y0) is a critical point of its graph for the projection parallel to the x-axis, with the same critical value y0.
The function f(x) = x2 + 2x + 3 is differentiable everywhere, with the derivative f ′(x) = 2x + 2. This function has a unique critical point −1, because it is the unique number x0 for which 2x0 + 2 = 0.