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Proving An Inequality

  1. Feb 13, 2016 #1
    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##?
     
  2. jcsd
  3. Feb 13, 2016 #2

    fresh_42

    Staff: Mentor

  4. Feb 13, 2016 #3
    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.
     
  5. Feb 13, 2016 #4

    fresh_42

    Staff: Mentor

    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.
     
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