Proving Inequality: Can Partial Derivatives Help?

[Physicist97]

Hello!
Say we have an inequality that says that ##f(x, y)>c## where ##f(x, y)## is a function of two variables and ##c## is a constant. Assume that we know this inequality to be true when ##x=a## and ##y=b##. If you show that the partial derivatives of ##f(x, y)## with respect to ##x## and ##y## are both greater than zero, does that prove that ##f(x, y)>c## whenever ##x## is greater than or equal to ##a## and ##y## is greater than or equal to ##b##?

 

[fresh_42]

Do you mean the partial derivatives are everywhere positive or just at ##(a,b)##?
For the latter I think this is a counter example:
https://en.wikipedia.org/wiki/Spont...g#/media/File:Mexican_hat_potential_polar.svg

Edit: If everywhere then cut-off the brim.

 

[Physicist97]

The partial derivatives are positive in the regions ##x>a## and ##y>b##. They could be positive everywhere, but the above is what I think is important to proving that inequality. I could be wrong, though.

 

[fresh_42]

The mexican hat potential is a counterexample. Only that the derivatives in (0,0) are zero. But then one could define a pole there.
All derivatives are positive, the function values let's say in a circle of radius r are all above c but not outside of it, i.e. for x,y > r.