Let $f(x)=\sin x, g(x)=2 x$ and $h(x)=\cos x$. If $\phi(x)=[go(f h)](x)$, then $\phi'' \left(\frac{\pi}{4}\right)$ is equal to
Answer & explanation
Correct answer: option 3
We have,
$(f h)(x)=f(x) h(x)=\sin x \cos x$
$\phi(x)=[go(f h)](x)=g((f h)(x))$
$\Rightarrow \phi(x)=g(\sin x \cos x)=2 \sin x \cos x=\sin 2 x$
$\Rightarrow \phi'(x)=2 \cos 2 x$
$\Rightarrow \phi''(x)=-4 \sin 2 x$
$\Rightarrow \phi''(\pi / 4)=-4 \sin \pi / 2=-4$