Question

Find intervals in which f(x) is increasing or decreasing
$f\left(x\right)=sinx(1+cosx),0\frac{\pi }{2}$

Answer 1

$f\left(x\right)=sinx(1+cosx)f\left(x\right)=sinx+\frac{2sinxcosx}{2}\Rightarrow f\left(x\right)=sinx+\frac{sin2x}{2}\therefore f^{\prime }\left(x\right)=cosx+cos2x{f}^{\prime }\left(x\right)=0\Rightarrow cosx+cos2x=0\Rightarrow cosx+2co{s}^{2}x-1=0\Rightarrow 2co{s}^{2}x+cosx-1=00\angle x\angle \frac{\pi }{2}2co{s}^{2}x+2cosx-cosx-1=0\Rightarrow 2cosx(cosx+1)-1(cosx+1)=0\Rightarrow \left(2\right(cosx-1\left)\right(cosx+1)=0$
$\Rightarrow cosx=\frac{1}{2}or-1$
$Now,since,0\angle x\angle \frac{\pi }{2}$
$\therefore cosx=\frac{1}{2}$
$\Rightarrow x=\frac{\pi }{3}$
Now, f(x) is increasing when, $f^{\prime }\left(x\right)\ge 0$
$\therefore cos\ge \frac{1}{2}\therefore xϵ[\frac{\pi }{2},\frac{\pi }{3}]$
and, f(x) is decreasing when, f'(x) < 0
$\therefore xϵ(\theta .\frac{\pi }{3})$