I need to determine all possible values for $a\in [0,2[$ such that the following holds for all $(x,y)\in\mathbb{R}^2$:
$$(2x+ay)(-x+6y)+(2y+ax)(-20y)\leq 0$$ or equivalent $$-2x^2-40y^2-21axy+12xy+a6y^2\leq 0$$
I've plotted the function and observed that it holds for all $a\in[0,\alpha[$ for some $\alpha$ between 1.3 and 1.4 and it doesn't hold for greater values. But I don't know how to prove it nor find such $\alpha$.
Say $$f(x)= -2x^2-40y^2-21axy+12xy+a6y^2$$ and since $f(x)\leq 0 $ for all $x$ we have $\Delta_x\leq 0$, so $$y^2(21a-12)^2+8y^2(6a-40)\leq 0$$ for all $y$, so
$$(21a-12)^2+8(6a-40)\leq 0$$
and solving this quadratic inequality you will get an answer.