Unfair coin tossing game,target,optimal fixed investment,

Question

Suppose the player has capital 1\$. He chooses a number $f\in[0,1]$.He tosses an unfair coin repeatedly, which wins for him, with probability $p$, a gain $q\times f \times$ current capital \$,where $q>1$ is a fixed number,say 2,and with probability $1-p$ a loss of $f \times$ current capital \$.What is his bold play,i.e. what is the optimal choice of $f$ if there are $M$ tosses? Second, what is the probability that he wins a target $T\$>1\$$ before $M$ rounds?