PYQ Vault
EbooksTutor

Constants that make a piecewise function differentiable everywhere

This type of question, asked 4 times: 2021 Paper 9, 8 Apr 2023, 30 Jan 2024, 24 Jan 2026 S1. All 5 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2021 Paper 9

    77
    If f(x)={1∣x∣,∣x∣≥1ax2+b,∣x∣<1f(x)=\begin{cases}\dfrac{1}{|x|}, & |x|\geq 1 \\ ax^{2}+b, & |x|<1\end{cases} is differentiable at every point of its domain, then the values of aa and bb are respectively:

    Tap an option to check your answer.

  2. Question 2

    8 Apr 2023

    172
    Let kk and mm be positive real numbers such that the function  f(x)={3x2+kx+1,0<x<1mx2+k2,x≥1\ f(x) =\left\{ \begin{matrix} 3x^{2}+ k\sqrt{x + 1}, & 0 < x < 1 \\ mx^{2}+k^{2}, & x \geq 1 \end{matrix} \right. is differentiable for all x>0x > 0. Then 8f′(8)f′(18)\frac{8f^{'}(8)}{f^{'}\left( \frac{1}{8} \right)} is equal to

  3. Question 3

    30 Jan 2024

    164
    Let aa and bb be real constants such that the function ff defined by f(x)={x2+3x+a,x≤1bx+2,x>1f(x) =\left\{ \begin{matrix} x^{2}+ 3x+a & ,x\leq 1 \\ bx+ 2 & ,x> 1 \end{matrix} \right. be differentiable on RR. Then, the value of ∫−22f(x)dx\int_{- 2}^{2} f(x)dx equals

    Tap an option to check your answer.

  4. Question 4

    30 Jan 2024

    89
    If the function f(x)={1∣x∣,∣x∣≥2ax2+2b,∣x∣<2f(x) =\left\{ \begin{matrix} \frac{1}{|x|} & ,|x| \geq 2 \\ ax^{2}+ 2b, & |x| < 2 \end{matrix} \right. is differentiable on RR, then 48(a+b)48(a + b) is equal to

  5. Question 5

    24 Jan 2026 S1

    66
    Let α,β∈R\alpha,\beta \in R be such that the function f(x)={2α(x2−2)+2βx,x<1(α+3)x+(α−β),x≥1f\left( x \right)=\left\{ \begin{matrix} 2\alpha\left( x^{2}- 2 \right)+ 2\beta x & ,x < 1 \\ \left( \alpha + 3 \right)x +\left( \alpha - \beta \right) & ,x \geq 1 \end{matrix} \right. be differentiable at all x∈Rx \in R. Then 34(α+β)34(\alpha + \beta) is equal to

    Tap an option to check your answer.