Find all critical points of the following function. $$ f(x, y)=x^{2}-4 x+y^{2}+4 y $$ What are the critical points? Select the correct choice below and fill in any answer boxes within your choice. A. The critical point(s) is/are (Type an ordered pair. Use a comma to separate answers as needed.) B. There are no critical points.

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We start by computing the partial derivatives of $$ f(x,y) = x^2 - 4x + y^2 + 4y $$.

1. The partial derivative with respect to $$ x $$ is  
   $$
   f_x(x, y) = \frac{\partial f}{\partial x} = 2x - 4.
   $$
   Setting $$ f_x(x,y) = 0 $$ gives  
   $$
   2x - 4 = 0 \quad \Rightarrow \quad x = 2.
   $$

2. The partial derivative with respect to $$ y $$ is  
   $$
   f_y(x, y) = \frac{\partial f}{\partial y} = 2y + 4.
   $$
   Setting $$ f_y(x,y) = 0 $$ gives  
   $$
   2y + 4 = 0 \quad \Rightarrow \quad y = -2.
   $$

Thus, the only critical point is  
$$
(2, -2).
$$

A. The critical point is $$(2, -2)$$.
The critical point is (2, -2).

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