Find extrema (local or global) point(s) of a function subject to the given constraint $$f(x, y) = x^2 + 4y^2 − 2x + 8y, x + 2y = 7.$$ Use Lagrange multiplier (gradient method).
Let the Lagrange multiplier be $$L(x,y,\lambda)=f(x, y)-\lambda g(x,y)$$ where $g(x,y)=x+2y-7.$
Then $\nabla L=0$ implies $\nabla f=\lambda \nabla g$, and $g(x,y)=0$.
We then get the system of equations $2x-2=\lambda, 4y+4=\lambda, x+2y=7$. Solving we get the stationary point as $(x,y)=(5,1)$.
Q1: How to check that there is a minimum at $(5,1)$?
Q2: How to check whether the extremum point is local or global extremum?