TITLE: Lagrange Multipliers Example
QUESTION [6 upvotes]: Minimize $$f(x,y) = x^2+y^2$$ subject to the constraint $xy=3$. 
I know the formula for Lagrange multipliers to be $\nabla f = \lambda \nabla g$ so we get a system of equations like this $$\begin{cases}f_x = \lambda g_x \\ f_y = \lambda g_y \end{cases}$$
However that gives me $2$ equations, but $3$ variables. I do not know what to do from here.

REPLY [2 votes]: Here's another approach for fun. We can use the AM-GM mean:
$$\sqrt{ab}\leq \frac{a+b}{2},\quad \text{for all}\ a,b\geq 0$$
where equality holds if and only if $a=b$. Use $a=x^2,\ b=y^2$ to get:
$$\sqrt{x^2y^2}\leq \frac{x^2+y^2}{2},$$
and our minimum will be where $x=y$. This implies $x=y=\pm\sqrt{3}$ and thus:
$$2\sqrt{(xy)^2} = 2|xy| = 6\leq x^2+y^2,$$
for all $x,y\in\mathbb{R}$ such that $xy=3$.