Final answer:
The critical points of the function f(x, y) = x² occur at all points where x equals zero, forming a line along the y-axis at x = 0.
Step-by-step explanation:
To find the critical points of a function f(x, y), you usually need to take the partial derivatives of the function with respect to x and y, then set them equal to zero and solve for the variables x and y. However, in this case, the function f(x, y) = x² does not depend on y, meaning that the partial derivative with respect to y will always be zero. Therefore, to find the critical points, we take the partial derivative with respect to x, which is 2x. We set 2x = 0 and solve for x, which gives us x = 0. As there is no dependency on y, all points of the form (0, y) are critical points for any real number y.