Question from insight

[8039]

I'm terrible at absolute value questions so just wondering how this one is solved. I'm asked to find the derivative of the function. I get to the part where it's cos(x) times sign(x) but don't fully know how they got the domain or how to show the working

Also, , find the maximum value of the function in the interval ... why is it ax-x^2 when outside of the absolute value sign

cheers  

[dopplereffect]

Alright, firstly it's best to draw the original graph.

Next, you have to recognise that abs(sin(x)) is + or - sin(x) when x is in certain domain/s.

In this graph, -sin(x) occurs for -π to 0 and from π to 2π. Sin(x) occurs from -2π to -π and from 0 to π.

Now turn the absolute function into a hybrid function with those restrictions in place, and then differentiate.

Then make sure that the derivative is not valid for points where the limit on both sides is not the same (for this graph the points at -π, 0 and π are not differentiable) so have to be excluded from the domain of the derivative function. Also any endpoints are also not differentiable because the limits on both sides cannot be found. Therefore the hybrid function will become that which is shown above.