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Least number of zeros of f' or f'' using Rolle's theorem

This type of question, asked 4 times: 2021 Paper 23, 29 Jun 2022, 28 Jul 2022, 30 Jan 2024. All 4 questions below, oldest first.

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  1. Question 1

    2021 Paper 23

    78
    Let ff be any continuous function on [0,2]\lbrack 0,2\rbrack and twice differentiable on (0,2)(0,2). If f(0)=0,f(1)=1f(0) = 0, f(1) = 1 and f(2)=2f(2) = 2, then

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  2. Question 2

    29 Jun 2022

    176
    Let ff and gg be twice differentiable even functions on (−2,2)( - 2,2) such that f(14)=0,f(12)=0,f(1)=1f\left( \frac{1}{4} \right)= 0,f\left( \frac{1}{2} \right)= 0,f(1) = 1 and g(34)=0,g(1)=2g\left( \frac{3}{4} \right)= 0,g(1) = 2 Then, the minimum number of solutions of f(x)g′′(x)+f′(x)g′(x)=0f(x)g^{''}(x) +f^{'}(x)g^{'}(x) = 0 in (−2,2)( - 2,2) is equal to

  3. Question 3

    28 Jul 2022

    83
    Let f:[0,1]→Rf:\lbrack 0,1\rbrack \rightarrow R be a twice differentiable function in (0,1)(0,1) such that f(0)=3f(0) = 3 and f(1)=5f(1) = 5. If the line y=2x+3y = 2x + 3 intersects the graph of ff at only two distinct points in (0,1)(0,1), then the least number of points x∈(0,1)x \in (0,1), at which f′′(x)=0f^{''}(x) = 0, is

  4. Question 4

    30 Jan 2024

    67
    Let g:R→Rg:R \rightarrow R be a non constant twice differentiable such that g′(12)=g′(32)g^{'}\left( \frac{1}{2} \right)=g^{'}\left( \frac{3}{2} \right). If a real valued function ff is defined as f(x)=12[ g(x)+g(2−x)]f(x) =\frac{1}{2}\lbrack\text{ }g(x) + g(2 - x)\rbrack, then

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