Suppose I have functions f and g; is there any theorem or technique that I can use to know if $f(x) \leq g(x)$ or if $f(x) \geq g(x)$ or neither in a given interval?
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I don't know if this is helpful and it doesn't solve your general problem, but as an example: if you have sat two real valued (differentiable) functions $f$ and $g$ defined on the same say closed interval $[a, b]$ of the real numbers, you could for example note that say $f(a) > g(a)$ for the left endpoint of the interval and then note that the derivative of $f$ is greater than or equal to the derivative of $g$ on the whole interval. If this is all so, then you would be able to conclude that $f(x) > g(x)$ for all x in the interval. But again, this approach would only apply to a small set of functions.
Another idea could be to (as above) first note that $f$ is greater than $g$ at some point in the interval and then show that $f(x) = g(x)$ does not have any solutions. Then $f$ clearly would be greater than $g$ on the whole interval.