Vasilev N.B.

A cherry, a sphere of radius $r$, is dropped into a round glass, the axial cross-section of which is the graph of the function $y=x^{4}$. For what largest $r$ will the sphere touch the lowest point of the bottom? (In other words, what is the maximum radius $r$ of a circle lying in the region $y \geq x^{4}$ and containing the origin?)