Answer
Maximum:$f(1,1)=e$, Minimum: $f(-1,1)=-e$
Work Step by Step
Our aim is to calculate the extreme values with the help of Lagrange Multipliers Method subject to the given constraints. For this, we have:$\nabla f(x,y)=\lambda \nabla g(x,y)$
This yields $\nabla f=\lt e^y,xe^y \gt$ and $\lambda \nabla g=\lambda \lt 2x,2y \gt$
From the given constraint condition $x^2+y^2=2$ we get, $e^y=\lambda 2x, xe^y=\lambda 2y$
Simplify. Thus, $x=\pm 1$
Since, $g(x,y)=x^2+y^2=2$ $\implies$ $y=-2,1$
When, $y=-2$, thus $x^2+y^2=2 \implies x^2=-2$; which is incorrect point.
Hence, Maximum value is $f(1,1)=e$, Minimum value is $f(-1,1)=-e$
