Question

lf $f\left(x\right)=\sin x+\cos x,0\le x\le 2\pi$ then $f$ is

• A. f is strictly increasing in $[0,\frac{\pi }{4})$
• B. f is strictly decreasing in $(\frac{\pi }{4}\frac{5\pi }{4})$
• C. f is strictly increasing in $(\frac{5\pi }{4},2\pi ]$
• D. None of these

Answer 1

Answer: (A), (B), (C)

f is strictly increasing in $[0,\frac{\pi }{4})$
f is strictly decreasing in $(\frac{\pi }{4}\frac{5\pi }{4})$
f is strictly increasing in $(\frac{5\pi }{4},2\pi ]$

$f\left(x\right)=\sin x+\cos x$
$f^{\prime }\left(x\right)=\cos x-\sin x$
$f^{\prime }\left(x\right)=0\Rightarrow \tan x=1$
$\Rightarrow x=\frac{\pi }{4},\frac{5\pi }{4}$
$f^{\prime }\left(x\right)>0$ for $x\in (0,\frac{\pi }{4})\cup (\frac{5\pi }{4},2\pi )$
Also, f'(x)<0 for $x\in (\frac{\pi }{4},\frac{5\pi }{4})$