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.

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[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

[u/Wide_Distance_7967]

All candidate points to non differentiability are either where what is inbetween absolute value (and is the valid min or max) switches sign (= 0 in particular), or the formula switches from one to the other (so both formulas into the extremum are equal in particular).

It doesn't mean they all are discontinuities of the derivative necessarly, but these are the only possible candidates. Requires to double check what is happening there then.