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$f(x)$ is a continuous function in $[0,1]$, and $f(x)>0$. Try to find all the functions $f(x)$ that satisfy $\int_0^1f(x)dx=1,\quad \int_0^1xf(x)=a,\quad \int_0^1x^2f(x)=a^2, $ where $a$ is a given number.

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    @RagibZaman sorry I make a mistake.2012-04-21

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Hint: Show that $\displaystyle\int_0^1 (x-a)^2f(x)\mathrm dx=0$. Then, $(x-a)^2f(x)\geqslant0$ for every $x$ in $(0,1)$, hence...