The method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function f (x, y),with (x,y) moving on a curve Γ with equation g (x, y) = 0, we should focus on (x0 , y0) ∈ Γ where two vector (fx (x0, y0) , fy (x0, y0)) and (gx (x0, y0) , gy (x0, y0)) parallel. Which means we need to find (x0, y0) ∈ Γ to exist real number λ to make (fx (x0, y0) , fy (x0, y0)) = λ (gx (x0, y0) , gy (x0, y0)).
-Explain me why we need to focus on finding (x0, y0) like that to find the local maxima and minima of a function f (x, y) on Γ.
-And can real number λ be zero? if yes, can you give me specific example?