Question

If $x_1,x_2,...x_n$ are n-positive real numbers then $\frac{x_1+x_2+...+x_n}{n}\ge \sqrt[n]{n{x}_1{x}_2....x_n}$, which is known as AM - GM inequality and the equality holds only when $x_1=x_2=x_3=...=x_n$.

$\Delta$ABC is acute angled triangle then the least value of tan A tan B tan C is

• A. $3\sqrt{3}$
• B. $\sqrt{3}$
• C. $\frac{1}{3\sqrt{3}}$
• D. $\frac{1}{\sqrt{3}}$

Answer 1

The correct option is A $3\sqrt{3}$

In $\Delta$ ABC tanA+ tanB + tanC= tanA tanB tanC and $cot\frac{A}{2}+cot\frac{B}{2}+cot\frac{C}{2}=cot\frac{A}{2}cot\frac{B}{2}cot\frac{C}{2}$
Use these these two results in AM and GM.