If l, m, n are d.c's of vector $\overline{\mathrm{OP}}$ then maximum value of lmn is

$$\begin{gathered} {\mathrm{l}}^2+{\mathrm{m}}^2+{\mathrm{n}}^2=1 \mathrm{A}.\mathrm{M}\ge \mathrm{G}.\mathrm{M} \\ \frac{{\mathrm{p}}^2+{\mathrm{m}}^2+{\mathrm{n}}^2}{3}=\sqrt[3]{{\mathrm{l}}^2{\mathrm{m}}^2{\mathrm{n}}^2} \\ {\mathrm{l}}^2{\mathrm{m}}^2{\mathrm{n}}^2 \le \frac{1}{27} \\ \mathrm{lmn} \le \frac{1}{3\sqrt{3}} \end{gathered}$$