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Equation linking f(x) to an integral of t f(t) up to x: differentiate, solve, find f at a point

This type of question, asked 4 times: 10 Apr 2023, 24 Jan 2025, 28 Jan 2025, 2 Apr 2025. All 4 questions below, oldest first.

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  1. Question 1

    10 Apr 2023

    65
    Let ff be a differentiable function such that x2f(x)−x=4∫0xtf(t)dt,f(1)=23x^{2}f(x) - x = 4\int_{0}^{x} tf(t)dt,f(1) =\frac{2}{3}. Then 18f(3)18f(3) is equal to

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  2. Question 2

    24 Jan 2025

    71
    Let f be a differentiable function such that (x+2)2f(x)−3(x+2)2=10∫0x(t+2)f(t)dt(x + 2)^{2}f(x) - 3(x + 2)^{2}= 10\int_{0}^{x} (t + 2)f(t)dt x≥0x \geq 0. Then f(2)f(2) is equal to _____\_\_\_\_\_ .

  3. Question 3

    28 Jan 2025

    66
    Let for some function y=f(x),∫0xtf(t)dt=x2f(x)y = f(x),\int_{0}^{x} tf(t)dt =x^{2}f(x), x>0x > 0 and f(2)=3f(2) = 3. Then f(6)f(6) is equal to :

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  4. Question 4

    2 Apr 2025

    127
    Let f:[1,∞)→[2,∞)f:\lbrack 1,\infty) \rightarrow \lbrack 2,\infty) be a differentiable function, If 10∫1xf(t)dt=5xf(x)−x5−910\int_{1}^{x} f(t)dt = 5xf(x) -x^{5}- 9 for all x≥1x \geq 1, then the value of f(3)f(3) is :

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