First slide

Extreme values and Periodicity of Trigonometric functions

Question

$The minimum value of 27^{\cos x} + 81^{\sin x} is equal to$

Moderate

A

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

$\frac{1}{3 \sqrt{3}}$
C

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

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

Solution

$\begin{array}{l} We have 27^{\cos x} + 81^{\sin x} \\ = 3^{3 \cos x} + 3^{4 \sin x} \geq 2 \cdot \sqrt{3^{3 \cos x} \cdot 3^{4 \sin x}} \\ = {2.3}^{\frac{1}{2} (3 \cos x + 4 \sin x)} \geq {2.3}^{- 5 / 2} = \frac{2}{9 \sqrt{3}} \end{array}$

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