Question

Assertion :If $A+B+C=\pi$, then the minimum value of $\prod \tan A$ is $3\sqrt{3}$.
$[\prod \tan A=\tan A+\tan B+\tan C]$ Reason: $A.M.\ge G.M.$

• A. Both the statements are TRUE and reason is the correct explanation of assertion.
• B. Both the statements are TRUE and reason is NOT the correct explanation of assertion.
• C. Assertion is TRUE and reason 2 is FALSE.
• D. Assertion is FALSE and reason is TRUE.

Answer 1

The correct option is A Both the statements are TRUE and reason is the correct explanation of assertion.

Using $A.M.\ge G.M.$

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

Also in a $△;\tan A+\tan B+\tan C=\tan A\cdot \tan B\cdot \tan C$

$\Rightarrow \frac{\tan A.\tan B.\tan C}{3}\ge {\left(\tan A\tan B\tan C\right)}^{1/3}$

Let $t=\tan A\cdot \tan B\cdot \tan C$

$\Rightarrow t\ge 3{\left(t\right)}^{1/3}$

Cubing both sides we get,

$t^3\ge 27t\Rightarrow t(t^2-27)\ge 0\Rightarrow t(t-3\sqrt{3})(t+3\sqrt{3})\ge 0$

$\therefore t\ge 3\sqrt{3}$

$\therefore$ Minimum value of $\prod \tan A$ is

$3\sqrt{3}$

$\Rightarrow$ Assertion is true and reason is correct explanation of assertion.