Inequality with assumptions

Question

I would like to find 2D region formed by $(a,c) \in \mathbb{R}^2$ such that the inequality $f(a,c,x,y) \geq 0$ holds for all $x,y \in [0,1]$. If I solve this inequality with reduce of the form

Reduce[f[a,c,x,y] >= 0 && 0 <= x <= 1 && 0 <= y <= 1, {a, c}, Reals]

the result uses the variables $x$ and $y$. Can you please help me how to input that the inequality should be true for all $x,y \in [0,1]$? Thanks in advance.

Answer 1

This can be done as follows.

ForAll[{x, y}, {x, y} \[Element] Rectangle[{0, 0}, {1, 1}],f (a, c, x, y) >= 0];
Resolve[%, {a, c}, Reals]

For example,

ForAll[{x, y}, {x, y} \[Element] Rectangle[{0, 0}, {1, 1}], c*x^2 + y^2 + a >= 0];
Resolve[%, {a, c}, Reals]

a >= 0 && c >= -a