Answer
Maximum:$f(0,-1)=f(0,1)=1$, Minimum: $f(-2,0)=f(2,0)=-4$
Work Step by Step
Use Lagrange Multipliers Method:
$\nabla f(x,y)=\lambda \nabla g(x,y)$
This yields $\nabla f=\lt y,x \gt$ and $\lambda \nabla g=\lambda \lt 8x,2y \gt$
Using the constraint condition $4x^2+y^2=8$ we get, $y=\lambda 8x, x=\lambda 2y$
After solving, we get $x=\pm 1$
Since, $g(x,y)=x^2+y^2=10$ gives $y=2$
Hence, Maximum:$f(0,-1)=f(0,1)=1$, Minimum: $f(-2,0)=f(2,0)=-4$
