here is first couple of steps, the last sentence, maximise over a to yield the result. how u do that

Example 6.5.2 (Perpetual American put) Find the value of a perpetual American put option on non-dividend-paying stock, that is a contract that the holder can choose to exercise at any time t in which case the payoff is (K-S). Solution(s): We sketch two possible solutions to this problem, first via the free boundary problem of Proposition 6.5.1 and second via the expectation price. Since the time to expiry of the contract is always infinite, V(t, x) is a function of x alone and the exercise boundary must be of the form Sf(t) = a for all t > 0 and some constant a. The option will be exercised as soon as S, 0. This means that the value will be of the form V (0, So) = E [e="ta (K a)+], where t = inf{t > 0: S, 0 and some constant a. The option will be exercised as soon as S, 0. This means that the value will be of the form V (0, So) = E [e="ta (K a)+], where t = inf{t > 0: S,