Question

For $\theta \in[0, \pi]$, let $f(\theta)=\sin (\cos \theta)$ and $g (\theta)=\cos (\sin \theta) .$ Let $a =\max _{0 \leq \theta \leq \pi} f (\theta), b =\min _{0 \leq \theta \leq \pi}$ $f (\theta), c =\max _{0 \leq \theta \leq \pi} g (\theta)$ and $d =\min _{0 \leq \theta \in \pi} g (\theta)$. The correct inequalities satisfied by $a , b , c , d$ are

• A. b< d < c < a
• B. d < b < a < c
• C. b< d < a < c
• D. b < a < d < c

Answer 1

Answer: (C)

$f(\theta)=\sin (\cos \theta)$
$g(\theta)=\cos (\sin \theta)$
$f'(\theta)=\cos (\cos \theta)(-\sin \theta)<0 \forall \theta \in[0, \pi]$
$\therefore f(\theta)$ decreases monotonically
$\therefore a =\max f(\theta)=f(0)=\sin 1$
$b =\min f(\theta)=f(\pi)=-\sin 1$
$g'(\theta)=-\sin (\sin \theta) \cos \theta$

$g(\theta)=1 ; g(\pi)=1 ; g\left(\frac{\pi}{2}\right)=\cos 1$
$\therefore c=\max g(\theta)=1$
$d=\min g(\theta)=\cos 1$
$\therefore b < d < a < c$