AHSME 1962 Problem 37

$ABCD$ is a square with side of unit length. Points $E$ and $F$ are taken respectively on sides $AB$ and $AD$ so that $AE = AF$ and the quadrilateral $CDFE$ has maximum area. In square units this maximum area is: $\textbf{(A)}\ \frac{1}2 \textbf{(B)}\ \frac{9}{16} \textbf{(C)}\ \frac{19}{32} \textbf{(D)}\ \frac{5}{8} \textbf{(E)}\ \frac{2}3$