Problem 8.1. The numbers $x, y, z$ are such that $x \in[-3,7], y \in[-2,5], z \in[-5,3]$.

(a) (1 point) Find the smallest possible value of the quantity $x^{2}+y^{2}$.

(b) (3 points) Find the smallest possible value of the quantity $x y z - z^{2}$.
# Answer:

(a) (1 point) 0.

(b) (3 points) -200.

Solution. (a) Note that $x^{2} \geqslant 0$ and $y^{2} \geqslant 0$, so $x^{2}+y^{2} \geqslant 0$. The value $x^{2}+y^{2}=0$ is possible when $x=0, y=0$.

(b) Note that $|x y z|=|x||y||z| \leqslant 7 \cdot 5 \cdot 5$ and $z^{2} \leqslant 5^{2}$, so $x y z-z^{2} \geqslant-7 \cdot 5 \cdot 5-5^{2}=-200$. The value $x y z-z^{2}=200$ is possible when $x=7, y=5, z=-5$.