Let y=f(x) for some function f, satisfy _0^x tf(t),dt = x^2f(x), x0, and f(2)=3.

Question

Let $y=f(x)$ for some function $f$, satisfy $\int_0^x tf(t)\,dt = x^2f(x)$, $x>0$, and $f(2)=3$. Then $f(6)$ is:

Accepted Answer

Differentiate: $xf(x) = x^2f'(x)+2xf(x) \implies -xf(x)=x^2f'(x) \implies \dfrac{f'(x)}{f(x)}=-\dfrac1x$.

$$\ln f(x) = -\ln x+\ln c \implies f(x)=\frac{c}{x}.$$

Using $f(2)=3 \Rightarrow c=6$. So $f(x)=6/x$, and $f(6)=1$.

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