Lagrange Multipliers

Constrained optimisation is the bread and butter of portfolio theory and econometrics. Lagrange multipliers turn a "maximise $f$ subject to $g=0$" problem into an unconstrained problem in one extra variable.

Find the maximum and minimum values of

$f(x,y)=x^2+y^2$

subject to the constraint

$g(x,y)=x^2+xy+y^2-3=0.$

State the points $(x,y)$ at which each extremum occurs.