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Points where a function built from |x| and |x - a| is not differentiable

This type of question, asked 3 times: 2 May 2023 S1, 10 May 2024 S2, 11 May 2024 S2. All 3 questions below, oldest first.

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  1. Question 1

    2 May 2023 S1

    120
    Let KK be the set of all real values of xx, where the function f(x)=sin⁡∣x∣−∣x∣+2(x−π)cos⁡∣x∣f(x) = \sin|x| - |x| + 2(x-\pi)\cos|x| is not differentiable. Then the set KK is

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  2. Question 2

    10 May 2024 S2

    123
    Let S={t∈R:f(x)=∣x−π∣(e∣x∣−1)sin⁡∣x∣S=\{t\in\mathbb{R}: f(x)=|x-\pi|(e^{|x|}-1)\sin|x| is not differentiable at t}t\}, then SS is

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  3. Question 3

    11 May 2024 S2

    115
    The set of all points, where the derivative of the function f(x)=x1+∣x∣f(x) = \frac{x}{1+|x|} exists, is

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