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Points where a max or min of several functions is not differentiable

This type of question, asked 6 times: 2021 Paper 4, 26 Jun 2022, 15 Apr 2023, 28 Jan 2025, 4 Apr 2025, 4 Apr 2026 S1. All 6 questions below, oldest first.

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  1. Question 1

    2021 Paper 4

    86
    A function ff is defined on [−3,3]\lbrack - 3,3\rbrack as
    f(x)={min⁡{∣x∣,2−x2},−2≤x≤2[∣x∣],2<∣x∣≤3f(x) =\left\{ \begin{matrix} \min\left\{ |x|,2 -x^{2} \right\}, & - 2 \leq x\leq 2 \\ \lbrack|x|\rbrack & ,2 < |x| \leq 3 \end{matrix} \right.
    where [x]\lbrack x\rbrack denotes the greatest integer ≤x\leq x. The number of points, where ff is not differentiable in (−3,3)( - 3,3) is

  2. Question 2

    26 Jun 2022

    155
    Let f(x)=min⁡{1,1+xsin⁡x},0≤x≤2πf(x) = \min\{ 1,1 + x\sin x\},0 \leq x \leq 2\pi. If mm is the number of points, where ff is not differentiable and nn is the number of points, where ff is not continuous, then the ordered pair (m,n)(m,n) is equal to

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  3. Question 3

    15 Apr 2023

    66
    Let [x]\lbrack x\rbrack denote the greatest integer function and f(x)=max⁡{1+x+[x],2+x,x+2[x]},0≤x≤2f(x) = \max\{ 1 + x + \lbrack x\rbrack,2 + x,x + 2\lbrack x\rbrack\},0 \leq x \leq 2. Let mm be the number of points in [0,2], where ff is not continuous and nn be the number of points in (0,2)(0,2), where ff is not differentiable. Then (m+n)2+2(m + n)^{2}+ 2 is equal to:

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  4. Question 4

    28 Jan 2025

    74
    Let
    f(x)={3x,x<0min⁡{1+x+[x],x+2[x]},0≤x≤25,x>2f(x) =\left\{ \begin{matrix} 3x, & x < 0 \\ \min\{ 1 + x + \lbrack x\rbrack,x + 2\lbrack x\rbrack\}, & 0 \leq x \leq 2 \\ 5, & x > 2 \end{matrix} \right.
    where [.] denotes greatest integer function. If α\alpha and β\beta are the number of points, where f is not continuous and is not differentiable, respectively, then α+β\alpha+\beta equals. ………\ldots\ldots\ldots

  5. Question 5

    4 Apr 2025

    75
    Let mm and nn be the number of points at which the function f(x)=f(x) = max⁡{x,x3,x5,……,x21},x∈R\max\left\{ x,x^{3},x^{5},\ldots\ldots,x^{21} \right\},x \in \mathbb{R}, is not differentiable and not continuous, respectively. Then m+nm + n is equal to:

  6. Question 6

    4 Apr 2026 S1

    75
    The number of points, at which the function f(x)=max{6x,2+3x2}+∣x−1∣cos⁡∣x2−14∣,x∈(−π,π)f(x) = max\left\{ 6x,2 + 3x^{2} \right\} + |x - 1|\cos\left| x^{2} - \frac{1}{4} \right|,x \in ( - \pi,\pi), is not differentiable, is ____\_\_\_\_ .