I need to find a constant $a$ such that for all $x_1 x_2 > 0$:
$a > - \frac{x_1^2 + 7 x_2^2}{2 x_1 x_2}$
that is to say the supremum of the term on the right hand side. My question is how to find it. Of course, $x_1 x_2 > 0$ holds when both variables have the same sign. So I tried to simplify my problem a bit and now I want to find a constant $b = -2 a$ such that:
$\frac{x_1}{x_2} + \frac{7 x_2}{x_1} > b$
Looking at both inequalities, I thought 0 is the solution, but this can't be right. Because if $x_1$ is very small, the other term becomes very large, so there has to be a better boundary. Using some techniques (positive definite matrices), a friend of mine arrived at:
$a > \sqrt{7}$
however, I don't know how to apply those techniques yet and I have to get the result without those techniques. Does anyone have an idea, how I can get the boundary easily? I don't know whether $\sqrt{7}$ is correct by the way.
Thanks in advance.
PS: This is not a "real" homework, it's just something I think I have to show in order to solve the whole problem.