Let F(x, Y) = $y e ^ { x }$ + X

Let f(x, y) = $y e ^ { x }$ + x $y ^ { 2 }$ . At which point does f(x, y) have a possible maximum or minimum value?

A) $\left( \frac { \sqrt { 2 } } { 2 } , - \frac { \sqrt { 2 } \mathrm { e } \sqrt { 2 } / 2 } { 2 } \right)$
B) (0, 0)
C) $\left( \frac { 1 } { 2 } , - \frac { \mathrm { e } ^ { 1 / 2 } } { 2 } \right)$
D) $\left( \frac { 1 } { 2 } , - \mathrm { e } ^ { 1 / 2 } \right)$
E) none of these