In single-variable real function, For example, if we want to prove $f(x)\geq g(x)$ for any non-negative real number $x,$ one can start by checking the first derivative $f(x)-g(x)$ is nonngeative or not (to check increasing-ness). Then, (maybe) by checking that $f(0)-g(0)$ is nonnegative or not.
My question, is there any (simple) method to prove like the above for two-variable function? i.e $$f(x,y)\geq g(x,y)$$ (for example) with condition $x,y\geq 0.$
Any help would be appreciated.
Thank you.
The analog of your example would be to check if $f(0,0)\geq 0$ and check the gradient $\nabla f(x,y)\cdot(x,y)\geq 0$ for every $(x,y)$. This will answer your question because of the following reasoning:
Let $g(t)=f(tx,ty)$. Then, by the fundamental theorem of calculus, $$f(x,y)=\int_0^1g'(t)dt +f(0,0).$$ Computing this derivative explicitly using the chain rule, you will get your answer. If you need further help, I can update my answer.