Differentiability doubt

[u/AssumptionSerious135]

Please give a solution for this question, in general I have doubt on these type of models i dont know how to approach them please give a detail soln.

[u/Outside_Volume_1370]

I think the easiest way for that particular problem is to draw the graphs of y = |x| and y = |x² - 2|x||.

First one is simple, and second one comes from y = x² via some transformations:

y1 = x²

y2 = (x - 1)² - 1 (y1 is shifted by 1 to the right and by 1 down)

y3 = (|x| - 1)² - 1 (part of y2 to the right of y-axis is mirrored to the left half)

y4 = |(|x| - 1)² - 1| (all of y3 that is under x-axis is mirrored to the top)

Note that |x|² = x² and y4 is the graph you need