I am trying to solve the following problem.
Minimize $$f(x,y) = x^2 + y^2$$
Subject to $$g(x,y) = ax^2 +2bxy +cy^2-1 = 0$$ $$(x,y)\in\mathbb{R^2}$$ Where $$a>0,c>0,2b>a+c$$ a) Use Lagrange theorem to show that $\exists$ $\lambda\in\mathbb{R}$ such that $Df(x^*,y^*) = \lambda Dg(x^*,y^*)$
This is simple because we have non degenerate constraint qualification .
b) use a) to show that $\lambda = f(x^*,y^*)$.
This is the part that I am stuck on I have written the Lagrangian and the FOC’s but I couldn’t figure out how to show this part, I tried inverting $Dg(x^*,y^*)$ in the equation in part a) but realized it is a vector and can’t (in general) be inverted, I also know that $f(x^*,y^*) >0$ because of the constraint, can someone please provide the solution .
IMPORTANT- Please note that I am not to just solve the FOC’s along with the constraint find $(x^*,y^*)$ and find the optimal value then show that result in b) holds, the question uses the result of part b) to find $f(x^*,y^*)$ directly without finding $(x^*,y^*)$
Suppose $A>0$ is symmetric. Let $f(x) = x^T x$ and $g(x) = x^TAx -1$, then if $x^*$ is a solution to $\min \{ f (x) | g(x) = 0 \}$ we have $g(x^*) = 0$ and $Df(x^*) = \lambda Dg(x^*)$. Expanding gives $x^*= \lambda Ax^*$ and so $(x^*)^T x^* = \lambda (x^*)^T Ax^* = \lambda$. Hence $f(x^*) = \lambda$.