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Find f when f(x) contains its own derivatives at fixed points

This type of question, asked 3 times: 13 May 2024 S2, 14 May 2024 S1, 26 Apr 2025 S2. All 3 questions below, oldest first.

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  1. Question 1

    13 May 2024 S2

    132
    Let f:R→Rf:R\to R be a function such that f(x)=x3+x2f′(1)+xf′′(2)+6, x∈Rf(x)=x^3+x^2f'(1)+xf''(2)+6,\ x\in R, then f(2)f(2) equals

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  2. Question 2

    14 May 2024 S1

    137
    Let f:R→Rf:R\rightarrow R be a function such that f(x)=x3+x2f′(1)+xf′′(2)+6f(x)=x^3+x^2f'(1)+xf''(2)+6, x∈Rx\in R, then f(2)f(2) equals

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  3. Question 3

    26 Apr 2025 S2

    149
    If f(x)=3x2+2xf′(1)+f′′(2)f(x) = 3x^{2}+ 2xf^{'}(1) +f^{''}(2), then f(x)=f(x) =... .

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