Answer
Minimum: $f(1,1)=f(-1,-1)=2$
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 $\nabla f(x,y)=\lt 2x,2y \gt$ and $\lambda \nabla g(x,y)=\lambda \lt y,x \gt$
As per the given constraint condition, we have $xy=1$ we get, $2x=\lambda y, 2y=\lambda x$
Simplify. Thus, $x=\pm 1$
Since, $g(x,y)=xy=1 \implies $$y=\pm 1$
We can see that the function has no maximum value, only possess minimum value $f(1,1)=f(-1,-1)=2$.
