We consider coefficient problems for functions in the classSof functions that are univalent in the unit disk and with the additional requirement that the derivative also be univalent in the disk. We restrict the problem further to consider only functions with real coefficients. We solve the problem of finding the maximum value of each of the even coefficientsa2a4.…a14in this restricted class. The extremal function is the same for each of these coefficients and it is given explicitly. The same function maximizes these coefficients within the larger class of typically real functions with typically real first derivative. This function definitely does not maximize the odd coefficients nor the even coefficientsa16a18.… within this larger class. The method of proof is to use sequences of polynomials from a certain restricted class of polynomials and to solve a related problem for the polynomials. These results can then be interpreted to solve the stated problem.