Answer
Maximum:$f(1,2)=f(-1,-2)=2$, Minimum: $f(-1,2)=f(1,-2)=-2$
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 $\nabla f=\lt y,x \gt$ and $\lambda \nabla g=\lambda \lt 8x,2y \gt$
From the given constraint condition $4x^2+y^2=8$ we get, $y=\lambda 8x, x=\lambda 2y$
After simplifications, we get $x=\pm 1$
Since, $g(x,y)=x^2+y^2=10$ $\implies$ $y=2$
Hence, Maximum value is $f(1,2)=f(-1,-2)=2$, Minimum value is $f(-1,2)=f(1,-2)=-2$
