Question

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

Answer 1

Correct Answer - B
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