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

Answer 1

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

Answer 2

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

Answer 3

i thought parts didn't seem to lead me anywhere

Answer 4

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?

Answer 5

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

Answer 6

Maybe if playing with series falls through... LOL

\[\int_0^1 x^2 f(x) dx = \int_0^1 \int_0^1 xyf(x)f(y)dxdy\]

Answer 7

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

Answer 8

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