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

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

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Maximum of $\cot A\cot B\cot C$ occurs at $A=B=C=\dfrac{\pi}{3}$ $K_{\max}=\cot^3\dfrac{\pi}{3}=\left(\dfrac{1}{\sqrt3}\right)^3=\dfrac{1}{3\sqrt3}$

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