Let $\rho(x)$ be a probability density function on $\mathbb{R}$ with prescribed variance $\sigma^2$, so that: $$\int_\mathbb{R} \rho(x)\, dx = 1$$ and $$\int_\mathbb{R} x^2 \rho(x), dx = \sigma^2$$ Fact: the density function which maximizes the entropy functional $$S(\rho) = -\int_\mathbb{R} \rho(x) \log \rho(x)\, dx$$ with the constraints above is the normal distribution $$\rho(x) = \frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{x^2}{2\sigma^2}}$$ This can be proved using basic techniques from the calculus of variations.

My question: can this be used to prove the central limit theorem? In other words, can one show directly that the limiting distribution of the average of a sequence of i.i.d. random variables maximizes entropy?

Actually, I don't care too much about entropy. I'm mainly interested in the possibility of a variational proof of the central limit theorem.