Differentiability of a function

[Rohit_]

Consider the function f(x)=|cos(x)|+|sin(2-x)|, At which point f is not differentiable?

1. {(2n+1)π/2, nϵZ}
2. {nπ, nϵZ}
3. {nπ+2, nϵZ}
4. {nπ/2, nϵZ}

I first thought of trying using graphical method, but it was not useful since I am getting graph of a this function because of combinations of absolute and trignometric fn and their sum seems typical, and other way I thought of splitting functions in different interval but not getting instead.
Anyone have any trick to do this question.

 

[Zermelo]

Well the |x| function usually proves to be problematic at x=0. If you break up your problem into cases where cosx > 0 (<0) and sin(2-x) >0 (<0), you will get some nice function that you can take the derivative of easily (there will be 4 of these cases, can you see why?). For example, if the cosine is positive and sine is negative, you will have cosx - sin(2-x), if you remember the definition of absolute value. So, the only problematic points are where cosine is (or sine is 0). To evaluate the (existance of the) derivative at these points, you should use the limit definition of the derivative, but be careful when approaching zero from the left side (h<0) and right side (h>0), because your function has absolute values. It really doesn’t make sense to do this if you haven’t evaluated the differentiability of |x| using this way, I don’t know what is expected of you, but this is the most correct way to do it (someone correct me if I’m wrong!)

 

[Deleted member 4993]

For these "complicated" plotting, I use pencil/paper and WolframAlpha , both and side-by-side, to save time.