if `Agt0,Bgt0 and A+B=pi/3` then the maximum value of tanA.tanB is (A) `1/3` (B) `1/6` (C) `1/2` (D) 1

Options

A. `(2)/(sqrt(3))`
B. `(2)/(3^(1//4))`
C. 1
D. `2sqrt(3)`

Answer

A

Explanation

`tanA tanB = tanA tan ((pi)/(3)-A)=f("say")`
`therefore" " f'(A)0"at" =(pi)/(6)"and"" "f_(+)^(') ((pi)/(6))lt0,f_(-)^(')((pi)/(6))gt0" " "or" " "tan(A+B)=(tanA+tanB)/(1-tanAtanB)="cons" tan t`
`therefore " "tanA+tanB` should be minimum
Also `tanA + tanB ge2tan((A+B)/(2))ge(2)/(sqrt(3))` (Using concavity)