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  $f:\left[0,\mathrm{\infty }\right)\to R$ be a continuous strictly increasing function, such that ${f}^{3}\left(x\right)={\int }_{0}^{x} t\cdot {f}^{2}\left(t\right)dt$ for every  Then value of f(6) is ___.

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### Solution:

Given ${f}^{3}\left(x\right)={\int }_{0}^{x} t\cdot {f}^{2}\left(t\right)dt$

Differentiating, $3{f}^{2}\left(x\right){f}^{\mathrm{\prime }}\left(x\right)=x{f}^{2}\left(x\right)$

But  $f\left(0\right)=0⇒C=0$

$f\left(6\right)=6$

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