Where does overbetting turn negative?

(a) For $b = 1$, compute the zero $f_0$ of $g$ numerically for $p = 0.55, 0.6, 0.7$ and compare with $2f^*$. (b) For a general bet $X$ with $m = E X$, $v = E X^2$, $m_3 = E X^3$, show that $g(f) = fm - \frac12 f^2v + \frac13 f^3m_3 + O(f^4)$, and use it to explain why the zero of $g$ lies near $2m/v$ for small edges and moves right when $m_3$ is large. (c) Verify (b) on the long-shot bet $p = 0.02$, $b = 100$: compute $m, v, m_3$, the cubic term at $f = 2m/v$, the exact $g(2f^*)$ and the exact zero $f_0$.