Question

If $f^{\prime }\left(\sin \theta \right)<0$ and $f^{\prime \prime }\left(\sin \theta \right)>0\forall \theta \in (0,\frac{\pi }{2})$ and $g\left(\theta \right)=f\left(\sin \theta \right)+f\left(\cos \theta \right),$ then

• A. $g\left(\theta \right)$ is decreasing in $x\in (\frac{\pi }{4},\frac{\pi }{2})$
• B. $g\left(\theta \right)$ is decreasing in $x\in (0,\frac{\pi }{4})$
  
• C. $g\left(\theta \right)$ is increasing in $x\in (\frac{\pi }{4},\frac{\pi }{2})$
• D. $g\left(\theta \right)$ in increasing in $x\in (0,\frac{\pi }{4})$

Answer 1

Answer: (B), (C)

The correct options are
B $g\left(\theta \right)$ is decreasing in $x\in (0,\frac{\pi }{4})$

C $g\left(\theta \right)$ is increasing in $x\in (\frac{\pi }{4},\frac{\pi }{2})$

To find; Monotonicity of $g\left(\theta \right)$ in $(0,\frac{\pi }{4})$ and $(\frac{\pi }{4},\frac{\pi }{2})$

$g\left(\theta \right)=f\left(\sin \theta \right)+f\left(\cos \theta \right)g^{\prime }\left(\theta \right)=f^{\prime }\left(\sin \theta \right)\times \cos \theta -f^{\prime }\left(\cos \theta \right)\times \sin \theta .$

$f^{\prime }\left(\sin \theta \right)<0,f^{\prime \prime }\left(\sin \theta \right)>0\forall \theta \in (0,\frac{\pi }{2})$

Let $x=\sin \theta ,f^{\prime }\left(x\right)<0\forall x\in (0,1)$

$\because \theta \in (0,\frac{\pi }{2})\Rightarrow \cos \theta \in (0,1)$
$f$ is decreasing in $(0,1)$

$f^{\prime \prime }\left(x\right)>0\forall x\in (0,1)$
$\Rightarrow f^{\prime }$ is increasing in $(0,1)$

$g^{\prime }\left(\theta \right)=f^{\prime }\left(\sin \theta \right)\cos \theta -f^{\prime }\left(\cos \theta \right)\sin \theta$
$f^{\prime }\left(\sin \theta \right)<0,f^{\prime }\left(\cos \theta \right)<0$

In $(0,\frac{\pi }{4})\Rightarrow \cos \theta >\sin \theta$
$f^{\prime }\left(\cos \theta \right)>f^{\prime }\left(\sin \theta \right)\text{and}{f}^{\prime }<0$
$\Rightarrow g^{\prime }\left(\theta \right)<0$
$\therefore g\left(\theta \right)$ is decreasing in $x\in (0,\frac{\pi }{4})$

In $(\frac{\pi }{4},\frac{\pi }{2})\Rightarrow \sin \theta >\cos \theta$
$f^{\prime }\left(\sin \theta \right)<f^{\prime }\left(\cos \theta \right)\text{and}{f}^{\prime }<0$
$\Rightarrow g^{\prime }\left(\theta \right)>0$
$g\left(\theta \right)$ is increasing in $x\in (\frac{\pi }{4},\frac{\pi }{2})$