11.2. Find the greatest and the least value of the function $y=(\arcsin x) \cdot(\arccos x)$.
Answer: The maximum value $=\pi^{2} / 16$ (at $x=\sqrt{2} / 2$), the minimum value $=-\pi^{2} / 2$ (at $x=-1$). Solution. The values of $\arcsin x$ and $\arccos x$ for any $x \in[-1 ; 1]$, as is known, are related by the equation $\arcsin x+\arccos x=\pi / 2$. Thus, we need to investigate the function $y(t)=t(\pi / 2-t)$, where $t=\arcsin x \in[-\pi / 2 ; \pi / 2]$. This quadratic function with a negative leading coefficient attains its maximum value at the point $t=\pi / 4$ (the vertex of the parabola), which is $\pi^{2} / 16$. The minimum value is attained at the boundary of the interval $[-\pi / 2 ; \pi / 2]$, specifically at the point $t=-\pi / 2$ and it is $-\pi^{2} / 2$ (at the other end of the interval, at $t=\pi / 2$, the value is zero). The corresponding values of $x$ at which the maximum and minimum values of the function are achieved are: $x=\sqrt{2} / 2$ and $x=-1$.