Answer
Maximum:$f(3,1)=10$, Minimum: $f(-3,-1)=-10$
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 $\nabla f(x,y)=\lt 3,1 \gt$ and $\lambda \nabla g(x,y)=\lambda \lt 2x,2y \gt$
From the given question, by using the constraint condition $x^2+y^2=10$ we get, $3=\lambda 2x, 1=\lambda 2y$
After simplifications, we get $x=\pm 3$
Since, $g(x,y)=x^2+y^2=10$ $\implies$ $y=\pm 1$
Hence, Maximum value is $f(3,1)=10$, Minimum value is $f(-3,-1)=-10$
