Question

The minimum value of $2cos\theta +\frac{1}{sin\theta }+\sqrt{2}tan\theta$ in the interval $(0,\frac{\pi }{2})$ is

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

Since $\theta \in (0,\frac{\pi }{2})\therefore sin\theta >0,cos\theta >0andtan\theta >0$

Now using AM $\ge$ GM property

$\frac{2cos\theta +\frac{1}{sin\theta }+\sqrt{2}tan\theta }{3}\ge {(2cos\theta \times \frac{1}{sin\theta }\times \sqrt{2}tan\theta )}^{\frac{1}{3}}$

$\Rightarrow$ $2cos\theta +\frac{1}{sin\theta }+\sqrt{2}tan\theta \ge 3(2\sqrt{2}{)}^{\frac{1}{3}}$

= 3${\left(\right(\sqrt{2}{)}^2)}^{\frac{1}{3}}=3\sqrt{2}$