If A, B, C are acute positive angles such tha | vedclass

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(A) Given $A + B + C = \pi$,we know that $	an A + 	an B + 	an C = 	an A 	an B 	an C$.Since $A, B, C$ are acute angles,$	an A, 	an B, 	an C >

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$K \le \frac{1}{3\sqrt{3}}$

Vedclass · Answer

(A) Given $A + B + C = \pi$,we know that $	an A + 	an B + 	an C = 	an A 	an B 	an C$.Since $A, B, C$ are acute angles,$	an A, 	an B, 	an C > 0$.By the Arithmetic Mean-Geometric Mean $(AM \ge GM)$ inequality:$\frac{	an A + 	an B + 	an C}{3} \ge (	an A 	an B 	an C)^{1/3}$.Substituting $	an A + 	an B + 	an C = 	an A 	an B 	an C$:$\frac{	an A 	an B 	an C}{3} \ge (	an A 	an B 	an C)^{1/3}$.Let $X = 	an A 	an B 	an C$. Then $\frac{X}{3} \ge X^{1/3} \Rightarrow X^{2/3} \ge 3 \Rightarrow X \ge 3^{3/2} = 3\sqrt{3}$.Since $K = \cot A \cot B \cot C = \frac{1}{	an A 	an B 	an C} = \frac{1}{X}$,we have $K = \frac{1}{X} \le \frac{1}{3\sqrt{3}}$.

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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