# Implicit function theorem find neighbourhood of the point

1. Feb 12, 2015

### gothloli

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 $\frac{\partial F(x,y)}{\partial y}$ >0. Find the largest possible r0 > 0 so that for all x, $\left | x -a \right |$ < 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. Feb 18, 2015