$f(x)$ 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.
The construction of a continuous function f(x)
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calculus
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4Hint: Integration by parts... – 2012-04-21
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2$f(x)>0 $ and $\int^1_0 f(x) dx = 0$ ?? Even if you meant $ f(x) \geq 0 $ we see that $f(x)=0$ identically is forced. Is this the right question? – 2012-04-21
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0@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...