Question

If $A,B,C$ are acute positive angles such that $A+B+C=\pi$ and $\cot A\cot B\cot C=k$, then

• A. $k\le \frac{1}{3\sqrt{3}}$
• B. $k>\frac{1}{3}$
• C. $k\ge \frac{1}{3\sqrt{3}}$
• D. $k<\frac{1}{9}$

Answer 1

The correct option is A $k\le \frac{1}{3\sqrt{3}}$

$A+B+C=\pi$ and $0<A,B,C<\frac{\pi }{2}$, using A.M. $\ge$ G.M. $\Rightarrow \frac{\cot A+\cot B+\cot C}{3}\ge k^{1/3}$
$\cot A\cot B\cot C=k\le \frac{1}{3\sqrt{3}}$ (the minimum value of A.M. is G.M. and in this case $A=B=C=\frac{\pi }{3}$)