Question

The ratio of the maximum and minimum attained by the function $f(x)=1+2\sin x+3{\cos }^{2}x,0\le x\le \frac{2\pi }{3}$ is

• A. $3:1$
• B. $13:9$
• C. $9:4$
• D. $8:13$

Answer 1

Answer: (B)

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 $\sin x=1x\in \left[0,\frac{2\pi }{3}\right]$
$\therefore$ Ratio of Maximum to Minimum $13:9$.