14. In a rectangular parallelepiped with a body diagonal of length $l$, find the maximum value of the sum of the projections of the body diagonal on the three adjacent faces.



The above text translated into English, please keep the original text's line breaks and format, and output the translation result directly.
14. Let the length, width, and height of a rectangular prism be $a$, $b$, and $c$, respectively, then $a^{2}+b^{2}+c^{2}=l^{2}$. It is easy to know that the sum of the projections of the diagonal of the rectangular prism on the adjacent three faces is $x=\sqrt{a^{2}+b^{2}}+\sqrt{b^{2}+c^{2}}+\sqrt{a^{2}+c^{2}}$.
By the Cauchy-Schwarz inequality, we have
$$
x^{2}=\left(1 \cdot \sqrt{a^{2}+b^{2}}+1 \cdot \sqrt{b^{2}+c^{2}}+1 \cdot \sqrt{a^{2}+c^{2}}\right)^{2} \leqslant\left(1^{2}+1^{2}+1^{2}\right)\left[\left(a^{2}+b^{2}\right)+\left(b^{2}+c^{2}\right)+\left(c^{2}+a^{2}\right)\right]=6 l^{2} \text {. }
$$

Therefore, $x_{\max }=\sqrt{6} l$