Task 9/83 Determine the maximum and minimum of the functions

$$
y=x_{1} x_{2} \pm \sqrt{\left(1-x_{1}\right)^{2}\left(1-x_{2}\right)^{2}}
$$

for $\left|x_{1}\right| \leq 1,\left|x_{2}\right| \leq 1$, without using tools of differential calculus!
Due to $\left|x_{1}\right| \leq 1,\left|x_{2}\right| \leq 1$, the substitutions $x_{1}=\sin \alpha, x_{2}=\sin \beta$ are possible. This results in

$$
y=\sin \alpha \sin \beta \pm \sqrt{\left(1-\sin ^{2} \alpha\right)\left(1-\sin ^{2} \beta\right)}=\sin \alpha \sin \beta \pm \cos \alpha \cos \beta= \pm \cos (\alpha \mp \beta)
$$

From the range of values of the cosine function, it follows that $-1 \leq y \leq 1$. For

$$
y=x_{1} x_{2}+\sqrt{\left(1-x_{1}^{2}\right)\left(1-x_{2}^{2}\right)}
$$

the maximum is achieved for $x_{1}=x_{2}$, and the minimum for $x_{1}=-x_{2}= \pm 1$. If the minus sign applies, then the maximum occurs at $x_{1}=x_{2}=-1$, and the minimum at $x_{1}=-x_{2}$.