The title is general, but what I am specifically interested in, is how to solve the following problem:
$$\text{Maximize } c $$ $$\text{Subject to:}$$ $$a+b+c<0$$ $$b^2-4ac<0$$ $$a,b \in \mathbb{R}, c \in \mathbb{Z}$$
I can solve it by first showing that $c$ cannot be positive or zero, and then showing that $c=-1$ has a solution. But this kind of solution is not satisfying to me, I want to just get the solution of $c = -1$ directly. How can I do it?
The condition $b^2 - 4ac < 0$ means that the quadratic form $$ f(x,y) = a x^2 + bxy + cy^2 $$ is definite, either positive definite or negative definite. Then $$ f(1,1) = a + b + c < 0$$ means negative definite, since it is a value attained by the form. Finally $$ f(0,1) = c, $$ so $c < 0.$
Since they want $c$ integral, $ c \leq -1.$