The least value of `6tan^2varphi+54cot^2varphi+18` is 54 when A.M.`geq` GM. Is applicable for `6tan^2varphi,54cot^2varphi,18` 54 when A.M.`geq` GM. Is applicable for `6tan^2varphi,54cot^2varphi,18` is added further 78 when `tan^2varphi=cot^2varphi` (I) is correct, (II) is false (I) and (II) are correct (III) is correct None of the above are correct

Options

A. (I) is correct, (II) is false
B. (I) and (II) are correct
C. (III) is correct
D. (III) is correct

Answer

B

Explanation

Applying A.M `ge` G.M in `6 tan^(2) phi, 54 cot^(2) phi, 18` we get
`(6 tan^(2) phi + 54 cot^(2) phi + 18)/(3) ge (6 xx 54 xx 18)^(1//3) ge 18`
Now equality holds when
`6 tan^(2) phi = 54 cot^(2) phi = 18`
`implies tan^(2) phi = 3` and `cot^(2) phi = (1)/(3)`
Hence, the statements I and II are correct