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Q.
$f(x)=$ maximum $\{2 \sin x, 1-\cos x\}$ is not differentiable when $x$ is equal to
Continuity and Differentiability
Solution:
$f(x)=\max \{2 \sin x, 1-\cos x\}$ 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 $x=\pi-\cos ^{-1}(3 / 5)$
$\forall x \in(0, \pi)$
