State how to find the stationary points of a $C^{1}$ function $f(x, y)$ restricted to the circle $x^{2}+y^{2}=1$, using the method of Lagrange multipliers. Explain why, in general, the method of Lagrange multipliers works, in the case where there is just one constraint.

Find the stationary points of $x^{4}+2 y^{3}$ restricted to the circle $x^{2}+y^{2}=1$.