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Implicit function theorem find neighbourhood of the point

  1. Feb 12, 2015 #1
    1. The problem statement, all variables and given/known data
    F(x,y) = y2 - x4. At the point a = (0.5, 0.25) the implicit function theorem holds. Find the largest r1neighbourhood of a s.t [itex] \frac{\partial F(x,y)}{\partial y} [/itex] >0. Find the largest possible r0 > 0 so that for all x, [itex]\left | x -a \right |[/itex] < r0 implies F(x, 0.25 - r1) < 0 and F(x, 0.25 + r1) >0
    2. Relevant equations
    implicit function theorem

    3. The attempt at a solution
    dyF(x,y) = 2y>0 means y>0, since only condition is that y>0 and 0.25 - r1 < y < 0.25 + r1, r1 = 0.25. But that wouldn't make sense since then F(x, 0.25 - 0.25) = 0 which wouldn't follow the implicit function theorem. My question is how can you find the largest possible values since r0 and r1 can be anything > 0.
  2. jcsd
  3. Feb 18, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
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