How to find the the minimum non-negative value of a function:
$$f(x,y)=ax^2+by^2+cx+dy+e$$
s.t. $x$ lies in $[0, A]$ and $y$ lies in $[A, \infty),$
where $A$ is a known constant.
or simply $0\leq x\leq A\leq y$
Example: $f(x,y)=-x^2 + 2y^2 + 3y +8$ has a minimum positive value of $12$ for $A=1$. I have found this graphically but I would like to find the solution analytically.
Any help would be beneficial.