I'm looking for critical points for the following function for which I know there are an infinite number:
$$ f(x, y) = x^2 + 4y^2 - 4xy + 2 $$
I get partial derivatives:
$$ f_x(x, y) = 2x - 4y $$ $$ f_y(x, y) = -4x + 8y $$
But when I set these equal to 0, I get an identity:
$$ -4x + 8y = 0 $$ $$ x = 2y $$
$$ 2x - 4y = 0 $$ $$ 2(2y) - 4y = 0 $$ $$ 0 = 0 $$
The two partial derivative are just linear combinations of each other. Does this along with the identity I get when deriving critical points in some way prove that the equation has infinitely many critical points?
There are infinitely many critical points because $(x,y)$ is a critical point if and only if $x=2y$. In other words$$\{\text{critical points of }f\}=\{(2y,y)\mid y\in\Bbb R\},$$which is an infinite set.