Number of possible functions 
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. 
 A: 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 $C$.
A: 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.
