Let $f(x)$ and $g(x,y)$ be two functions.
I would like to prove that
$$\frac{g(x,y)}{f(x)}\geq 0$$
If I want to do so, would it be sufficient if I prove it for the biggest value of $f(x)$?
I mean, let $f(x)\leq M$ for all $x$.
If I prove
$$\frac{g(M,y)}{M}\geq 0$$
would that be sufficient?
Thank you.
On
For this case, the actual max and min of $f$ are less important that its sign (So if $f>0$ that's important, $f> -3$ does not help so much). Again $f \leqslant -1$ is nice to know if true, as then all you need to do is find when $g(x, y)\leqslant 0$.
In general study when $f(x)$ and $g(x, y)$ share the same sign, if necessary considering different parts of the domains separately. Which portions of the domains to consider are often given by the zeros of these functions.
No. Suppose $f(x)=1$ for all $x$. The condition $g(1,y) \geq 0$ for all $y$ does not guarantee that $g(x,y) \geq 0$ for all $x,y$. I will let you write down a counterexample.