Kelly Exceeds Full Investment

A favorable bet has limited downside: staking a fraction $f$ of wealth, you gain the full amount $f$ with probability $p=0.7$ but lose only half the stake, $0.5f$, with probability $0.3$. (a) Find the growth-optimal fraction $f^*$. (b) If you cannot borrow (so $f\le 1$, i.e. you can stake at most all your wealth), what fraction do you actually bet?