By definition the region inside the parabola $y = x^{2}$ is the set of points $(a,b)$ such that $b \geq a^{2}$. We are interested in those circles all of whose points are in this region. A bubble at a point $P$ on the graph of $y = x^{2}$ is the largest such circle that contains $P$. (You may assume the fact that there is a unique such circle at any given point on the parabola.) (a) A bubble at some point on the parabola has radius 1. Find the center of this bubble. (b) Find the radius of the smallest possible bubble at some point on the parabola. Justify.
By definition the region inside the parabola $y = x^{2}$ is the set of points $(a,b)$ such that $b \geq a^{2}$. We are interested in those circles all of whose points are in this region. A bubble at a point $P$ on the graph of $y = x^{2}$ is the largest such circle that contains $P$. (You may assume the fact that there is a unique such circle at any given point on the parabola.)
(a) A bubble at some point on the parabola has radius 1. Find the center of this bubble.
(b) Find the radius of the smallest possible bubble at some point on the parabola. Justify.