Question

Assertion :(A): If $A>0$ & $B>0,A+B=\frac{\pi }{6}$, then the maximum value of $\tan A\tan B$ is $7-4\sqrt{2}$ Reason: (R): If $x_1+x_2+x_3+.......x_n=\lambda$(constant), then value of $x_1,x_2,x_3,.......x_n$ is greatest when $x_1=x_2=x_3=.......x_n$

• 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. Both Assertion flase but Reason is true

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Assertion:

$A+B=\frac{\pi }{6}$

$\tan (A+B)=\tan \frac{\pi }{6}$

$\frac{\tan A+\tan B}{1-\tan A\tan B}=\frac{1}{\sqrt{3}}$

$1-\tan A\tan B=\sqrt{3}(\tan A+\tan B)$

$\tan A\tan B=1-\sqrt{3}(\tan A+\tan B)$

So, $\tan A\tan B$ is maximum when $\tan A+\tan B$ is minimum.

And this is only possible when $\tan A=\tan B=\frac{\pi }{12}$

Hence,

$max\left(\tan A\tan B\right)={\tan }^2\frac{\pi }{12}=(2-\sqrt{3}{)}^2=7-4\sqrt{3}$

Reason is true

Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion.