Mathematics - Application of Derivative Question with Solution | TestHub

Let $$\begin{array}{rl}& f(\theta )=\frac{1}{{\sin }^2\theta +3\sin \theta \cos \theta +5{\cos }^2\theta }\\ & \text{ Again let }\\ & g(\theta )={\sin }^2\theta +3\sin \theta \cos \theta +5{\cos }^2\theta \\ & =\frac{1-\cos 2\theta }{2}+5\left(\frac{1+\cos 2\theta }{2}\right)+\frac{3}{2}\sin 2\theta \\ & =3+2\cos 2\theta +\frac{3}{2}\sin 2\theta \\ & \therefore g(\theta {)}_{\min }=3-\sqrt{4+\frac{9}{4}}=3-\frac{5}{2}=\frac{1}{2}\\ & \therefore f(\theta )=\frac{1}{g(\theta {)}_{\min }}=2\end{array}$$