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Q.
The ratio of the maximum and minimum attained by the function $f(x)=1+2 \sin x+3 \cos ^{2} x, 0 \leq x \leq \frac{2 \pi}{3}$ is
TS EAMCET 2020
Solution:
Given, $f(x)=1+2 \sin x+3 \cos ^{2} x$
$f(x)=1+2 \sin x+3-3 \sin ^{2} x$
$f(x)=4-3\left(\sin ^{2} x-\frac{2}{3} \sin x+\frac{1}{9}\right)+\frac{1}{3}$
$f(x)=\frac{13}{3}-3\left(\sin x-\frac{1}{3}\right)^{2}$
Maximum value of $f(x)=\frac{13}{3}$ at $\sin x=\frac{1}{3}$
Minimum value of $f(x)=\frac{9}{3}$ at $\sin x=1 x \in\left[0, \frac{2 \pi}{3}\right]$
$\therefore $ Ratio of Maximum to Minimum $13: 9$.