Question

Find the greatest & the least values of the following functions in the given interval, if they exist. $y=\sin 2x-x[-\frac{\pi }{2},\frac{\pi }{2}]$

• A. $\frac{\pi }{2}$ and $-\frac{\pi }{2}$
• B. $\pi$ and $-\pi$
• C. $\frac{3\pi }{2}$ and $-\frac{3\pi }{2}$
• D. None of the above

Answer 1

The correct option is A $\frac{\pi }{2}$ and $-\frac{\pi }{2}$

Given, $y=\sin 2x$

$y^{\prime }\left(x\right)=2\cos 2x-1=0$
$\cos 2x=\frac{1}{2}$
$x=-\pm \frac{\pi }{6}$
Hence, $y_{x=\frac{\pi }{6}}$ $=\frac{\sqrt{3}}{2}-\frac{\pi }{6}$. ...(i)
And $y_{x=-\frac{\pi }{6}}$ $=-\frac{\sqrt{3}}{2}+\frac{\pi }{6}$. ...(ii)
Now, $f\left(\frac{\pi }{2}\right)=0-\frac{\pi }{2}<ii$
And $f(-\frac{\pi }{2})=\frac{\pi }{2}>i$
Hence greatest value is $\frac{\pi }{2}$
and Least value is $-\frac{\pi }{2}$.