Application Of Derivatives Ques 122
122. The maximum value of the expression $\frac{1}{\sin ^{2} \theta+3 \sin \theta \cos \theta+5 \cos ^{2} \theta}$ is ……
(2010)
Show Answer
Answer:
Correct Answer: 122.(2)
Solution:
Formula:
Increasing and decreasing of a function:
- Let
$ f(\theta)=\frac{1}{\sin ^{2} \theta+3 \sin \theta \cos \theta+5 \cos ^{2} \theta} $
Again let, $g(\theta)=\sin ^{2} \theta+3 \sin \theta \cos \theta+5 \cos ^{2} \theta$
$ \begin{aligned} & =\frac{1-\cos 2 \theta}{2}+5 (\frac{1+\cos 2 \theta}{2})+\frac{3}{2} \sin 2 \theta \\ & =3+2 \cos 2 \theta+\frac{3}{2} \sin 2 \theta \\ \therefore \quad g(\theta)_{\min } & =3-\sqrt{4+\frac{9}{4}} \\ & =3-\frac{5}{2}=\frac{1}{2} \end{aligned} $
$\therefore \quad$ Maximum value of $f(\theta)=\frac{1}{1 / 2}=2$
BETA
