The number of possible continuous functions $f(x)$ defined on $[0,1]$ for which $I_1 = \int_0^1 f(x)dx=1$, $I_2 = \int_0^1 xf(x)dx= a$ , $I_3=\int_0^1 x^2 f(x)dx= a^2$ is/ are?
I have seriously no idea how to attempt this problem and find the number of functions. I was trying some integration by parts for I2, I3 but that really didn't help.
On
There is an infinite number of functions $g$ such that: $$\int_0^1 g(x) dx=\int_0^1 x g(x) dx=\int_0^1 x^2 g(x) dx=0$$ (for example orthogonal polynomials) and if $f$ is a solution and $g$ such that all the above integral are $0$ then $f+g$ is also $0$.
So the result is either $0$ or an infinite number of functions.
To show that there exist at least one solution you can consider: \begin{align} \phi: \mathbb R_2[X] \to \mathbb R^3\\ P \mapsto \left(\int_0^1 P(x) dx, \int_0^1 x P(x)dx , \int_0^1 x^2P(x) dx\right) \end{align} and use any linear algebra method (for example write the matrix) to show that this linear application is bijective.
Hint. Note that if there is one such $f$ then also $f(x)+tP(x)$ works for any real $t$ where $$P(x)=3(x-1/2)-20(x-1/2)^3$$ because $\int_0^1 P(x)dx=\int_0^1 xP(x)dx=\int_0^1 x^2P(x)dx=0$.
In order to find one example of $f$, consider a quadratic polynomial $A+Bx+Cx^3$, then solve the system $$\begin{cases} \int_0^1 f(x) dx=1\\ \int_0^1 xf(x) dx=a\\ \int_0^1 x^2f(x) dx=a^2 \end{cases}$$ which is the linear with respect to $A$, $B$ and $C$.