Minimum value of 27cosx + 81sinx is equal to

Foundation

Mathematics Foundation

Law of Exponents

Introduction to Trigonometry

Maximum and Minimum of Quadratic

Question

Minimum value of 27^cosx + 81^sinx is equal to

$\frac{2}{9 \sqrt{3}}$

$\frac{4}{3 \sqrt{3}}$

None of these

$\frac{2}{3 \sqrt{3}}$

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We know that
A.M. $\geq$ G.M.
$∴ \frac{27^{cosx} + 81^{sinx}}{2} \geq \sqrt{27^{cosx} \times 81^{sinx}}$
$\Rightarrow 27^{cosx} + 81^{sinx} \geq 2 \sqrt{3^{3 cosx + 4 sinx}}$
$\Rightarrow 27^{cosx} + 81^{sinx} \geq 2 \times \sqrt{3^{- 5}}$ $[∵ - 5 \leq 3 cosx + 4 sinx \leq 5]$
$\Rightarrow 27^{cosx} + 81^{sinx} \geq \frac{2}{9 \sqrt{3}}$
Hence, the minimum value of $27^{cosx} + 81^{sinx} is \frac{2}{9 \sqrt{3}}$