Question

The inequality $2^{\sin \theta }+2^{\cos \theta }\ge 2^{1-\left(1/\sqrt{2}\right)}$ holds for all real values of $\theta$.

• A. True
• B. False

Answer 1

The correct option is A True

Ans. $True$. We have $\frac{1}{2}[2^{\sin \theta }+2^{\cos \theta }]\ge \sqrt{2^{\sin \theta }.2^{\sin \theta }}$
$[\because A.M\ge G.M.]$
$\Rightarrow 2^{\sin \theta }+2^{\cos \theta }\ge {2.2}^{(\sin \theta +\cos \theta )/2}$ $\left(1\right)$
Now $(\sin \theta +\cos \theta )=\sqrt{2}\sin (\theta +\pi /4)\ge -\sqrt{2}$ for all real $\theta$.
$\therefore 2^{\sin \theta }+2^{\cos \theta }\ge {2.2}^{(\sin \theta +\cos \theta )/2}>{2.2}^{-\sqrt{2}/2}=2^{1-\left(1/\sqrt{2}\right)}$
Thus $2^{\sin \theta }+2^{\cos \theta }\ge 2^{1-\left(1/\sqrt{2}\right)}$.