The Implicit function theorem for a single equation in our textbook states that let F(x,y) be a function of class $C^1$ on some neighborhood of a point (a,b), for each x in the ball |x-a|< $r_0$ there is a unique y such that |y-b| < $r_1$
Given the function F(x,y)= ($y^2$- $x^4$)
at the point a= (0.5,0.25), the conditions of the IFT hold, and we are asked to find the largest $r_1$ such that $\partial_{y}$ F(x,y)>0 for all the points in the $r_1$ neighbourhood of the point a
I know that $\partial_{y}$=2y and for 2y>0, we can get y>0 and thus the inequality |y-0.25|<$r_1$, now I have restrict y to a bound but how should I find the limit for $r_1$?