Let  f:[0,∞)→R be a continuous strictly increasing function, such that f3(x)=∫0x t⋅f2(t)dt for every  x≥0 Then value of f(6) is ___.

Let  f:[0,)R be a continuous strictly increasing function, such that f3(x)=0xtf2(t)dt for every  x0 Then value of f(6) is ___.

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    Solution:

    Given f3(x)=0xtf2(t)dt

    Differentiating, 3f2(x)f(x)=xf2(x)

    f(x)0 f(x)=x3f(x)=x26+C

    But  f(0)=0C=0

    f(6)=6

     

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