If $f (x) = \frac{2 {sin}^{2} x + 2 sin x + 3}{{sin}^{2} x + sin x + 1}$ , then the number of integers in the range of $f (x)$ is

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Solution

$f (x) = \frac{2 {sin}^{2} x + 2 sin x + 3}{{sin}^{2} x + sin x + 1}$
$= 2 + \frac{1}{{sin}^{2} x + sin x + 1}$
$= 2 + \frac{1}{{(sin x + \frac{1}{2})}^{2} + \frac{3}{4}}$
$f (x)_{min} = 2 + \frac{1}{\frac{9}{4} + \frac{3}{4}} = \frac{7}{3}$
$f (x)_{max} = 2 + \frac{1}{3 / 4} = \frac{10}{3}$

Hence, range of $f (x)$ is $[\frac{7}{3}, \frac{10}{3}]$
$\Rightarrow$ Only one integral value i.e., $3$ .

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If f(x)=2sin2x+2sinx+3sin2x+sinx+1, then the number of integers in the range of f(x) is

f(x)=2sin2x+2sinx+3sin2x+sinx+1 =2+1sin2x+sinx+1 =2+1(sinx+12)2+34 f(x)min=2+194+34=73 f(x)max=2+13/4=103 Hence, range of f(x) is [73,103] ⇒ Only one integral value i.e., 3.

If $f (x) = \frac{2 {sin}^{2} x + 2 sin x + 3}{{sin}^{2} x + sin x + 1}$ , then the number of integers in the range of $f (x)$ is