Question

Assertion :If $A+B+C=\pi$, then minimum value of $\tan A\tan B\tan C$ is $3\sqrt{3}$ Reason: $AM\ge GM$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is flase but Reason is true

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Assertion:
Minimum area of triangle is when is equilateral triangle i.e
$A=B=C=\frac{\pi }{3}$
Therefore, $min\left(\tan A\tan B\tan C\right)={\left(\tan \frac{\pi }{3}\right)}^3={\left(\sqrt{3}\right)}^3=3\sqrt{3}$

Reason:
$A+B+C=\pi \Rightarrow A+B=\pi -C\tan (A+B)=\tan (\pi -C)$

$\Rightarrow \frac{\tan A+\tan B}{1-\tan A\tan B}=-\tan C$

$\Rightarrow \tan A+\tan B+\tan C=\tan A\tan B\tan C$ ...(1)

Applying A.M$\ge$ G.M on $\tan A,\tan B,\tan C$, we get

$\frac{\tan A+\tan B+t\tan C}{3}\ge \sqrt[3]{3\tan A\tan B\tan C}$

Substituting value from (1)
${\left(\tan A\tan B\tan C\right)}^{\frac{2}{3}}\ge 3\tan A\tan B\tan C\ge 3\sqrt{3}$