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Let `f:[0,oo) to R` be a continuous strictly increasing function, such that `f^3(x)=int_0^x tdotf^2(t)dt` for every `xgeq0.` Then value of `f(6)` is_______

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Verified by Experts

The correct Answer is:
6
Given `f^(3)(x)=int_(0)^(x)=int_(0)^(x)t.f^(2)(t)dt`
Differentiating `3f^(2)(x)f'(x)=xf^(2)(x)`
`f(x)!=0`
`:.f'(x)=x/3:.f(x)=(x^(2))/6+3`
But `f(0)=0impliesC=0`
`f(6)=6`
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