If A > 0, B > 0, and A + B = `pi/3` then the maximum value of tan A tan B is

Answer

3

Explanation

`A+b= pi//3`
`rArr tan (A+B) = sqrt3`
`rArr (tanA + tanB)/(1-tan A tan B) = sqrt3`
`rArr = (tan A + (y)/(tan A)) /( 1-y) = sqrt3`
` rArr tan^(2)A+ sqrt3 (y-1) tan A +y =0 `
For real values of `tan A`,
`" " 3(y-1)^(2) - 4y =0`
`rArr 3y^(2) - 10y + 3 ge 0`
`rArr (y-3)(3y-1) ge 0`
`rArr y le (1)/(3) or y ge 3`
But A, B `gt` 0 and A +B `= pi//3`
`rArr A, B lt pi//3`
`rArr tan A tan B lt 3` so, ` ge 3` is not a possibility.
Therefore, `y le (1)/(3)` i.e., max. value of `y ` is `1//3`