$f(x)=|cosx|+|sin(2−x)|$ At which of the following point $f$ is not differentiable?

  1. $(2n+1)\pi/2$
  2. $n\pi$
  3. $n\pi+2$
  4. $n\pi/2$

where $n \in \mathbb{Z}$.

When I tried to solve it graphically, I got that $|\sin x|$ is not differentiable at $n\pi$ where $n\in\mathbb{Z}$ and $|\cos x|$ is not differentiable at $(n+1)\pi/2$ (I am not sure about this one). So the given function must not be differentiable at option (1) and option (3). Am I correct?

(I know this is duplicate but from there I could not get the answer satisfactorily and I can not even comment so I am posting it here)

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    $\begingroup$ Your solution seems correct. What do you expect from an answer? $\endgroup$ – Aleksejs Fomins Mar 20 at 7:53
  • $\begingroup$ Actually I am not sure about the differentiability of $|cosx|$ and I am also confused among the options so I expect a concrete proof of the correct answers. $\endgroup$ – Huny Mar 20 at 7:56
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    $\begingroup$ $cos(x) = sin(x + \pi / 2)$, so it's differentiability is shifted by $\pi/2$ wrt $sin(x)$ $\endgroup$ – Aleksejs Fomins Mar 20 at 8:00
  • $\begingroup$ Got it!Thanks!! $\endgroup$ – Huny Mar 20 at 8:02

The values at which the function is potentially non-differentiable is where the individual terms are non-differentiable, i.e. where the argument of the absolute values change sign. These are the odd multiples of half-Pi, and the multiples of Pi plus two.

As Pi is irrational, these points never coincide, so that the "non-differentiabilities" cannot cancel each other.

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  • $\begingroup$ Actually I could not understand ""non differentiabilities can not cancel each other". How do they cancel and how can they not? (Actually I am preparing for an exam all by myself and with the help of internet so I think I am having problem in basics, I will appreciate your help) $\endgroup$ – Huny Mar 20 at 8:03
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    $\begingroup$ @Huny: consider the case of $2|\sin x|-|\sin 2x|$. $\endgroup$ – Yves Daoust Mar 20 at 8:04
  • $\begingroup$ Thanks!! Now I got it. $\endgroup$ – Huny Mar 20 at 8:06
  • $\begingroup$ Also I am confused for the fourth option. $|cosx|$ is not differentiable at the odd multiples of $\pi/2$ and here n can be both(even or odd) so how can we assure the differentiability of the function at this point? $\endgroup$ – Huny Mar 20 at 8:10
  • $\begingroup$ @Huny: the question is indeed ambiguous. $\endgroup$ – Yves Daoust Mar 20 at 8:49

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