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Consider the function $f(\theta)=|\cos\theta|+|\sin(2-\theta)|$
At which of the following points is $f$ not differentiable ?

1.$\{(2n+1)\frac{\pi}{2} : n \in Z\}$
2.$\{n\pi :n \in Z\}$
3.$\{n\pi+2 : n \in Z\}$
4.$\{\frac{n\pi}{2} :n \in Z\}$

How to solve this problem, please help

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Hint: At what points the function $x \mapsto |x|$ is differentiable? – levap Oct 29 '12 at 15:34
@ levap :x is not diff at o. option 1 and 3 are correct. am i right? – bdas Oct 29 '12 at 16:36
Yes, you are correct. – levap Oct 29 '12 at 18:12
up vote 1 down vote accepted

Hint: The prime answer is from @Dennis, but I think the points in which $|\cos(t)|$ is not differentiable are the points in which $\cos(t)=0$. Do the same for $|\sin(2-t)|$.

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Nice, Babak! I hope you awaken refreshed! ;-) – amWhy Apr 1 '13 at 0:10

Hints: First, consider $|x|$. At which points is this function differentiable and at which it isn't?
Now, remember that if we have two functions $f(x),g(x)$ such that $g(x)$ is differentiable at $x=a$ and $f(x)$ is differentiable at $x=g(a)$ then $f(g(x))$ is differentiable at $x=a$. (and $(f(g(x)))'|_{x=a}=g'(a)f'(g(a))$)

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