hey need some company to do this problem
find all continuous positive functions for 0

Question

hey need some company to do this problem
find all continuous positive functions for 0<=x<=1, and 
$$\int_0^1 f(x)dx=1$$
$$\int_0^1 xf(x)dx=\alpha$$
$$\int_0^1 x^2f(x)dx=\alpha^2$$
alpha is a real number

xapproachesinfinity · Answer

my first impression 
do this 
$$\int_0^1(x^2-x-1)f(x)=\alpha^2-\alpha-1$$

anonymous · Answer

i'm stumped
my first idea was parts, but that doesn't seem to help

xapproachesinfinity · Answer

i thought parts didn't seem to lead me anywhere

kainui · Answer

That's clever, it looks like you're headed towards some kind of fibonacci thing @xapproachesinfinity I was also thinking about IBP.

Have you tried taylor series?

xapproachesinfinity · Answer

i was looking into something of that nature @Kainui 
series didn't click to me

kainui · Answer

Maybe if playing with series falls through... LOL

$$\int_0^1 x^2 f(x) dx = \int_0^1 \int_0^1 xyf(x)f(y)dxdy$$

xapproachesinfinity · Answer

gotta go for now, i will work on it more tomorrow

kainui · Answer

Yeah good luck, this is an interesting problem I'd like to finish it.