Find symbolic expression for critical point

Question

Let $g(x,y)=f(x,y)+x+y$ and $f(x,y)=ax^2+by^2+cxy$ where $a>0$ and $4ab−c^2>0$

I need to find a symbolic expression the minimum of $g(x,y)$ as a function of $a,b,c$ (not $x,y$) and I figure I can do it by setting the gradient to zero:

$ \nabla g(x,y)=(2ax+cy+1,2by+cx+1)=0 \Rightarrow 2ax+cy+1=0 \Rightarrow x=\frac{-cy-1}{2a}. $

Then I get

$ 2by+c(\frac{-cy+1}{2a})=0 \Rightarrow y=\frac{ac}{4b-ac^2} $

Is this correct? It seems very ugly. Where do I go from here?

Answer 1

Let $\nabla g(x,y)=0$, namely $$ 2ax+cy+1=0,2by+cx+1=0.$$ From the first, one has $$ x=-\frac{cy+1}{2a}$$ and then put it in the second equation to get $$ 2by-c\frac{cy+1}{2a}+1=0. $$ So $$ y=\frac{-2a+c}{4ab-c^2}, x=\frac{-2b+c}{4ab-c^2}. $$

Answer 2

By Cramer,

$$x=\frac{\begin{vmatrix}-1&c\\-1&2b\end{vmatrix}}{\begin{vmatrix}2a&c\\c&2b\end{vmatrix}},y=\frac{\begin{vmatrix}2a&-1\\c&-1\end{vmatrix}}{\begin{vmatrix}2a&c\\c&2b\end{vmatrix}}.$$