Determine the value of $$\inf_{f \in C[0,1]} \int_0^1 t^2 \mathrm{Re}f(t)dt$$ subject to the conditions $\int_0^1f(t)dt = \int_0^1 tf(t)dt=0$ and $\int_0^1|f(t)|^2dt=1$. Is the infimum attained? If so, for which $f\in C[0,1]$?
Here, $C[0,1]$ is the set of complex-valued continuous functions on $[0,1]$.
I realize that the integrals in the question can be written in terms of the inner product $$\langle f,g \rangle = \int_0^1f(t)\overline{g(t)}dt.$$ If we do this, then the conditions say that $f$ is normalized and is orthogonal to the constant function at $1$ and the identity function. I have tried to write the inner product in the infimum in terms of the ones where the value is given but failed. I have also tried creating some some sort of orthonormal system, maybe using the Gram-Schmidt orthonormalization process, and somehow using the properties of an orthonormal system to evaluate the infimum, but have also not succeeded.