3. We will denote $\max (A, B, C)$ as the greatest of the numbers $A, B, C$. Find the smallest value of the quantity $\max \left(x^{2}+|y|,(x+2)^{2}+|y|, x^{2}+|y-1|\right)$.
Answer: 1.5.

Solution: Note that when $x^{2}>(x+2)^{2}$ for $x-1$. If $x=-1$, then both of these values are equal to 1, otherwise one of them is greater than 1. Similarly, for $|y|$ and $|y-1|$, when $y=1 / 2$ they are both equal to $1 / 2$, and when $y \neq 1 / 2$ one of them is greater. Therefore, the minimum is achieved at $x=-1, y=1 / 2$.