Question

$f(x)=$ maximum $\{2\sin x,1-\cos x\}$ is not differentiable when $x$ is equal to

• A. 1
• B. -1
• C. 0
• D. $\pi -{\cos }^{-1}\left(\frac{3}{5}\right)$

Answer 1

Answer: (D)

$f(x)=\max \{2\sin x,1-\cos x\}$ can be plotted as shown in the figure.

Thus, $f(x)=$ maximum $\{2\sin x,1-\cos x\}$ is not differentiable,
when $2\sin x=1-\cos x$
or $4{\sin }^{2}x=(1-\cos x{)}^2$ or $4(1+\cos x)=(1-\cos x)$
or $4+4\cos x=1-\cos x$ or $\cos x=-3/5$ or
$x={\cos }^{-1}(-3/5)$
Therefore, $f(x)$ is not differentiable at $x=\pi -{\cos }^{-1}(3/5)$
$\forall x\in (0,\pi )$