All the question is in the following image and there is no additional detail elsewhere. Thank you!

An investor with a strictly concave and increasing utility function u for final wealth wants to invest in a stock. The current price of the stock is p and the price tomorrow will be PA > p with probability or pe

0, 2 p 2 p 2p so an investment in the stock is a better than fair bet. (a) Write the optimization problem of the investor and the corresponding first-order conditions for optimality. (b) Assume u(W) = log(W), p = 1, pa = 2 and pe = 1/2. How many shares should the investor buy