Question

If A=$\{xϵR/\frac{\pi }{4}\le x\le \frac{\pi }{3}\}$ and $f\left(x\right)=\sin x-x$, then $f\left(A\right)$=

• A. $[\frac{\sqrt{3}}{2}-\frac{\pi }{3},\frac{1}{\sqrt{2}}-\frac{\pi }{4}]$
• B. $[-\frac{1}{\sqrt{2}}-\frac{\pi }{4},\frac{\sqrt{3}}{2}-\frac{\pi }{3}]$
• C. $[-\frac{\pi }{3},\frac{-\pi }{4}]$
• D. $[\frac{\pi }{4},\frac{\pi }{3}]$

Answer 1

Answer: (A)

$[\frac{\sqrt{3}}{2}-\frac{\pi }{3},\frac{1}{\sqrt{2}}-\frac{\pi }{4}]$

$f\left(x\right)=\sin x-x$Which is decreasing function in $[\frac{\pi }{4},\frac{\pi }{3}]$
$\therefore \frac{\pi }{4}\le x\le \frac{\pi }{3}$
$\Rightarrow f\left(\frac{\pi }{4}\right)\ge f\left(x\right)\ge f\left(\frac{\pi }{3}\right)$
$\Rightarrow \frac{1}{\sqrt{2}}-\frac{\pi }{4}\ge f\left(x\right)\ge \frac{\sqrt{3}}{2}-\frac{\pi }{3}$
$f\left(A\right)ϵ[\frac{\sqrt{3}}{2}-\frac{\pi }{3},\frac{1}{\sqrt{2}}-\frac{\pi }{4}]$