Question

Statement-1: Minimum value of function $f(\theta )={\sin }^4\theta +\frac{\sec \theta }{8\tan \theta }$ is $\frac{5}{16}$ for $\frac{\pi }{2}<\theta <\pi$. Let $a_i>0\forall i\in N$, then $\frac{a_1+a_2+a_3+\dots \dots ..+a_n}{n}\ge \sqrt[n]{a_1{a}_2{a}_3\dots \dots a_n}$
Statement-2: where equality is obtained at $a_1=a_2=a_3\dots \dots \dots =a_n$.

• A. Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.
• B. Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.
• C. Statement-1 is true, statement-2 is false.
• D. Statement-1 is false, statement-2 is true. .

Answer 1

Answer: (A)

Given $f(\theta )={\sin }^4\theta +\frac{\sec \theta }{8\tan \theta }={\sin }^4\theta +\frac{\mathrm{cosec}\theta }{8}={\sin }^4\theta +\frac{1}{8\sin \theta }$ As ${\sin }^4\theta ,\frac{1}{32\sin \theta },\frac{1}{32\sin \theta },\frac{1}{32\sin \theta },\frac{1}{32\sin \theta }$ are positive in $\left(\frac{\pi }{2},\pi \right)$
Now using A.M. $\ge$ G.M., we get
$\frac{{\sin }^4\theta +\frac{1}{8\sin \theta }}{5}\ge {\left(\frac{1}{(32{)}^4}\right)}^{1/5}\text{ or }{\sin }^4\theta +\frac{1}{8}\mathrm{cosec}\theta \ge \frac{5}{16}$
Hence minimum value of ${\sin }^4\theta +\frac{1}{8}\mathrm{cosec}\theta$ is $\frac{5}{16}$