• # question_answer In AABC with usual notations the least value $\frac{{{e}^{A}}}{A}+\frac{{{e}^{B}}}{B}+\frac{{{e}^{C}}}{C}$ is A)  $\frac{\pi }{3}\,{{e}^{\pi /9}}$                 B)  $\frac{\pi }{9}\,{{e}^{\pi /3}}$ C) $\frac{9}{\pi }\,{{e}^{\pi /3}}$                  D)  $\frac{3}{\pi }\,{{e}^{\pi /9}}$

We have, $\because$     $AM\ge GM$ $\frac{{{e}^{A}}}{A}+\frac{{{e}^{B}}}{B}+\frac{{{e}^{C}}}{C}\ge 3\,{{\left( \frac{{{e}^{A+B+C}}}{ABC} \right)}^{1/3}}$ $=3\,{{\left( \frac{{{e}^{\pi }}}{ABC} \right)}^{1/3}}$            ?(i) and $A+B+C\ge 3\,\,{{(ABC)}^{1/3}}$ $\Rightarrow$            $\frac{\pi }{3}\ge \,{{(ABC)}^{1/3}}$ $\Rightarrow$            $\frac{\pi }{3}\le \frac{1}{{{(ABC)}^{1/3}}}$ $\Rightarrow$            $\frac{3}{\pi }\,{{({{e}^{\pi }})}^{1/3}}\le {{\left( \frac{{{e}^{\pi }}}{ABC} \right)}^{1/3}}$ $\Rightarrow$            $3\left( \frac{3}{\pi } \right)\,{{e}^{\pi }}^{/3}\le 3{{\left( \frac{{{e}^{\pi }}}{ABC} \right)}^{1/3}}$ $\Rightarrow$            $\frac{{{e}^{A}}}{A}+\frac{{{e}^{B}}}{B}+\frac{{{e}^{C}}}{C}\ge 3\cdot \,\left( \frac{3}{\pi } \right){{e}^{\pi /3}}=\frac{9}{\pi }\,{{e}^{\pi /3}}$ [from Eq. (i)] $\Rightarrow$            Least vlaue is $\frac{9}{\pi }\,{{e}^{\pi /3}}$.