Answer
Maximum: $f(-2,\sqrt{12}),f(-2,-\sqrt{12})=47 $ and Minimum: $f(1,0)=-7$
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 $\nabla f(x,y)=\lt 4x-4,6y \gt$ and $\lambda g(x,y)= \lt x,y \gt$
Using the constraint condition $x^2+y^2 \leq 16$ we get, $x^2=\pm \dfrac{9}{2}$ and $x= \pm \dfrac{3}{\sqrt 2}$
$\nabla f(x,y)=\lt 4x-4,6y \gt$ and $\lambda g(x,y)= \lt x,y \gt$ we get $x=-2$
and $y=\pm \sqrt{12}$
Hence, Maximum: $f(-2,\sqrt{12}),f(-2,-\sqrt{12})=47 $ and Minimum: $f(1,0)=-7$
