Question

Minimum value of 27^{cos x} 81^{sin x}
1. -5
2. 1/5
3. 1/243
4. 1/27

Answer 1

Correct Answer - Option 3 : 1/243

27^{cos x} 81^{sin x}

3^{3 cos x} 3^{4 sin x}

⇒ 3^{(3 cos x + 4 sin x)}

Now,

3 cos x + 4 sin x can be written as:

\(\sqrt {{{\left( 3 \right)}^2} + {{\left( 4 \right)}^2}} \sin \left( {x + {{\tan }^{ - 1}}\frac{3}{4}} \right)\) 

\(= 5\sin \left( {x + {{\tan }^{ - 1}}\frac{3}{4}} \right)\)      ---(1)

[∵ a sin x + b cos x = R sin (x + θ)

Where \(R = \sqrt {{a^2} + {b^2}} ,\;\;θ = \left. {{{\tan }^{ - 1}}\frac{b}{a}} \right]\)

∴ The minimum value of (1) is (-5)

∴ The minimum value of

3^{(3 cos x + 4 sin x)} ⇒ 3^-5

\(= \frac{1}{{{3^5}}} = \frac{1}{{243}}\)