CSIR JUNE 2011 PART C QUESTION 61 SOLUTION (Differentiability of $f(x) = \mid cos \,x\mid + \mid sin(2-x)\mid$)

Consider the function $f(x) = \mid cos \,x\mid + \mid sin(2-x)\mid$. At which of the following points is $f$ is not differentiable.
1) $\{(2n+1)\frac{\pi}{2} : n \in \Bbb Z\}$,
2)$ \{n\pi : n \in \Bbb Z\}$,
3)$\{n\pi+2 : n \in \Bbb Z\}$,
4)$\{\frac{n\pi}{2}:n \in \Bbb Z\}$.
Solution:
We have $\mid x \mid$ is differentiable at $a$ if and only if $a \ne 0$. So the function $f(x)$ is not differentiable whenever $cos \,x$ or $sin \,(2-x)$ is zero. From the graph of $cos \,x$ we see that the zeros are $\{(2n+1)\frac{\pi}{2} : n \in \Bbb Z\}$. Next, for $sin \,x$ the zeros are given by $\{n\pi : n \in \Bbb Z\}$. Hence the zeros of $sin\,(2-x)$ are given by $2-x = n\pi$ which is equal to $\{2-n\pi : n \in \Bbb Z\} = \{2+n\pi : n \in \Bbb Z\}$. option(1) and (3) are correct.
Clearly, on the sets given in option (2) and (4), both the functions $cox \,x$ and $sin(2-x)$ does not vanish. So at that points the function $f$ will be differentiable. 

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