Question

The range of the function $f\left(x\right)=co{s}^2\frac{x}{4}+sin\frac{x}{4},xϵR$ is

• A. $[0,\frac{5}{4}]$
• B. $[1,\frac{5}{4}]$
• C. $(-1,\frac{5}{4})$
• D. $[-1,\frac{5}{4}]$

Answer 1

The correct option is D $[-1,\frac{5}{4}]$

$f\left(x\right)=1-si{n}^2\frac{x}{4}+sin\frac{x}{4}=-\{si{n}^2\frac{x}{4}-sin\frac{x}{4}\}+1=-\{{(sin\frac{x}{4}-\frac{1}{2})}^2-\frac{1}{4}\}+1$
$=\frac{5}{4}-{(sin\frac{x}{4}-\frac{1}{2})}^2$
Maximun $f\left(x\right)=\frac{5}{4}$
Minimum $f\left(x\right)=\frac{5}{4}-{(-1-\frac{1}{2})}^2=\frac{5}{4}-\frac{9}{4}=-1$
Range of $f\left(x\right)=[-1,\frac{5}{4}]$