Why Half-Kelly Keeps Three-Quarters of the Growth

For a small-edge repeated bet the expected log-growth is well approximated by the quadratic $G(f)\approx \mu f-\tfrac12\sigma^2 f^2$, where $\mu$ and $\sigma^2$ are the per-round mean and variance of the bet's return. Using this approximation, find the optimal fraction $f^*$ and show what fraction of the maximal growth $G(f^*)$ is retained by betting half-Kelly, $f=f^*/2$.