Question

Minimum value of 27 tan²x + 3cot²x is
1. 9
2. 18
3. 27
4. 30

Answer 1

Correct Answer - Option 2 : 18

Concept:

AM, GM, HM Formulas:

If A is the arithmetic mean 

⇔ \({\rm{A}} = \frac{{{\rm{a\;}} + {\rm{\;b}}}}{2}\)

If G is the geometric mean 

⇔ \({\rm{G}} = \sqrt {{\rm{ab}}} \)

Relation between AM, and GM

AM  ≥  GM

Calculation:

Given that,

27 tan2x + 3cot2x

Let a = 27tan2 x, b = 3cot2 x

We know that,

AM  ≥  GM

\(⇒\ \frac{27tan^2 x + 3cot^2 x}{2}\ ≥ \ \sqrt{27tan^2 x\times 3cot^2 x}\)

We know that, tan θ × cot θ = 1 

\(⇒\ {27tan^2 x + 3cot^2 x}\ ≥ \ 2\times 9\)

⇒ 27 tan2x + 3cot2x ≥ 18

⇒ 27 tan2x + 3cot2x ∈ [18, ∞) 

Hence, the minimum value is 18.