Question

In $△ABC$, least value of $\frac{e^A}{A}+\frac{e^B}{B}+\frac{e^C}{C}$ is equal to

• A. $\frac{9}{\pi }{e}^{\pi /3}$
• B. $\frac{\pi }{3}{e}^{\pi /3}$
• C. $\frac{\pi }{9}{e}^{\pi /3}$
• D. None of these

Answer 1

The correct option is A $\frac{9}{\pi }{e}^{\pi /3}$

We have
$\frac{e^A}{A}+\frac{e^B}{B}+\frac{e^C}{C}\ge 3{\left(\frac{e^{A+B+C}}{ABC}\right)}^{1/3}$ .....(Using $AM\ge GM$)

$\Rightarrow \frac{e^A}{A}+\frac{e^B}{B}+\frac{e^C}{C}\ge 3{\left(\frac{e^{\pi }}{ABC}\right)}^{1/3}$

Also $A+B+C\ge 3{\left(ABC\right)}^{1/3}$

$\Rightarrow \frac{\pi }{3}\ge {\left(ABC\right)}^{1/3}$

$\Rightarrow {\left(\frac{1}{ABC}\right)}^{1/2}\ge \frac{3}{11}$

$\Rightarrow 3{\left(\frac{e^{\pi }}{ABC}\right)}^{1/3}\ge \frac{9}{\pi }{e}^{\pi /3}$

$\Rightarrow \frac{e^A}{A}+\frac{e^B}{B}+\frac{e^C}{C}\ge \frac{9}{\pi }{e}^{\pi /3}$