Completely stumped as to how to solve this problem -_- We'd appreciate any help at all:
Suppose $f(x, y) = 2x^2 + 2xy + y^2$ and $g(x, y) = x^2 + y^2$
Show that minimizing $f$ with a constraint on $g$ yields points that also can be found by maximizing $g$ with a constraint on $f$. (In mean variance, you can minimize risk with a constraint on expected return, or you can maximize expected return with a constraint on risk)
There are a couple of ways to attack this.
1) To solve the constrained extremum problem using Lagrange multipliers, you would set $\mathrm d(f-\lambda g)=0$, which together with $g=\text{const.}$ gives you three equations in three unknowns. Compare that with what you get when you try to minimize $g$ while $f=\text{const.}$ using the same method.
2) The specific functions that appear in this problem make it equivalent to finding the direction in which the quadratic form $f(x,y)=2x^2+2xy+y^2$ takes on its maximum value. Going in the other direction (i.e., minimize $g$), you’re trying to find the closest point of a level curve of the quadratic form $f$ (which is an ellipse) to the origin, which amounts to determining the ellipse’s minor axis.