Question

For $ x \epsilon R , f (x) = | \log 2 - \sin x|$ and $g(x) = f(f(x))$, then :

• A. $g$ is not differentiable at $x = 0$
• B. $g'(0)$ = $cos(log2)$
• C. $g'(0)$ = $-cos(log2)$
• D. $g$ is differentiable at $x = 0$ and $g'(0) = -sin(log2)$

Answer 1

Answer: (B)

$g(x)=\left|\log _{e} 2-\sin \left(\left|\log _{e} 2-\sin x\right|\right)\right|$
At $x=0, g(x)=\log _{e}(2)-\sin \left(\log _{e} 2-\sin x\right)$
$\therefore g'(x)=\cos \left(\log _{e}(2)-\sin x\right) \times \cos (x)$
$\Rightarrow g'(0)=\cos \left(\log _{e}(2)\right)$