(Shrewd Prisoners Dilemma) Because of a prisoners constant

QUESTION:

(Shrewd Prisoners Dilemma) Because of a prisoners constant supplication, the king grants him this favor: He is given 2N balls, which differ from each other only in that half of them are green and half are red. The king instructs the prisoner to divide the balls between two identical urns. One of the urns will then be selected at random, and the prisoner will be asked to choose a ball at random from the urn chosen. If the ball turns out to be green, the prisoner will be freed. How should he distribute the balls in the urn to maximize his chances of freedom? Hint: Let g be the number of green balls and r be the number of red balls in the first urn. The corresponding numbers in the second urn are N g and N r. The probability that a green ball is drawn is f (g, r): f (g, r) = 1 2 * g g + r + N g 2N g r , . Find the maximum of this function of two variables (r and g). Note that the maximum need not occur at an interior point of the domain.