Answer
Maximum:$f(2,2)=e^4$, Minimum value does not exist.
Work Step by Step
Use Lagrange Multipliers Method:
$\nabla f(x,y)=\lambda \nabla g(x,y)$
This yields $\nabla f=\lt ye^{xy},xe^{xy} \gt$ and $\lambda \nabla g=\lambda \lt 3x^2,3y^2 \gt$
After solving, we get $x=y$
Since, $g(x,y)=x^3+y^3=16$ gives $x=2$
Thus, $x=y=2$
Hence, Maximum:$f(2,2)=e^4$. Minimum value does not exist.