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Polynomial defined using its own derivatives at fixed points

This type of question, asked 4 times: 24 Jan 2023, 1 Feb 2023, 27 Jan 2024, 21 Jan 2026 S2. All 4 questions below, oldest first.

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  1. Question 1

    24 Jan 2023

    151
    If f(x)=x3−x2f′(1)+xf′′(2)−f′′′(3), x∈Rf(x)=x^{3}-x^{2}f'(1)+xf''(2)-f'''(3),\ x\in R, then

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  2. Question 2

    1 Feb 2023

    86
    If f(x)=x2+g′(1)x+g′′(2)f(x) =x^{2}+g^{'}(1)x +g^{''}(2) and g(x)=f(1)x2+xf′(x)+f′′(x)g(x) = f(1)x^{2}+ xf^{'}(x) +f^{''}(x), then the value of f(4)−f(4) - g(4)g(4) is equal to

  3. Question 3

    27 Jan 2024

    88
    Let f(x)=x3+x2f′(1)+xf′′(2)+f′′′(3),x∈Rf(x) =x^{3}+x^{2}f^{'}(1) + xf^{''}(2) + f^{'''}(3), x \in R. Then f′(10)f^{'}(10) is equal to

  4. Question 4

    21 Jan 2026 S2

    53
    Let f(x)=x3+x2f′(1)+2xf′′(2)+f′′′(3)f(x) =x^{3}+x^{2}f^{'}(1) + 2xf^{''}(2) +f^{'''}(3), x∈Rx \in R. Then the value of f′(5)f^{'}(5) is :

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