11.18. The height of a regular quadrilateral prism $A B C D A_{1} B_{1} C_{1} D_{1}$ is half the length of the side of the base. Find the maximum value of the angle $A_{1} M C_{1}$, where $M$ is a point on the edge $A B$.
11.18. Let $A A_{1}=1, A M=x$. Introduce a coordinate system, the axes of which are parallel to the edges of the prism. The vectors $\overrightarrow{M A}_{\text {}}$ and $\overrightarrow{M C}_{1}$ have coordinates $(0,1,-x)$ and ( $2,1,2-x$ ); their scalar product is equal to $1-2 x+x^{2}=(1-x)^{2} \geqslant 0$. Therefore, $\angle A_{1} M C_{1} \leqslant 90^{\circ}$; when $x=1$ this angle is $90^{\circ}$.