In the problem
$f(x,y) = xy$ and $g(x, y) = x^2 + 9y^2 = 18$
I get
$y = 2λx$, $x = 18λy$ and $x^2 + 9y^2 = 18$ (the constraint).
All is fine, but I feel like I'll get two different answers depending on what my first step is.
I could do $y = 2λ(18λy) $ or I could do $x = 18λ(2λx)$
Almost identical. But...
$y = 36λ^2y$, $y - 36λ^2y = 0$, $y(1-36λ^2) y = 0$ or $λ = \pm1/6$.
or
$x = 36λ^2x$, $x - 36λ^2x = 0$, $x(1-36λ^2) = 0$ where $x = 0$ or $λ =\pm 1/6$.
Which component is zero flips!
What am I doing wrong?