Any local maxima or minima of the function z = f(x, y) subject to the constraint g(x, y) = 0 will be among those points (x0, y0) for which (x0, y0, λ0) is a solution of the system:
Fx(x, y, λ) = 0
Fy(x, y, λ) = 0
Fλ(x, y, λ) = 0
where F(x, y, λ) = f(x, y) + λg(x, y), provided that all the partial derivatives exist.
Maximize (or minimize) z = f(x, y)subject tog(x, y) = 0
F(x, y, λ) = f(x, y) + λg(x, y)
Fx(x, y, λ) = 0
Fy(x, y, λ) = 0
Fλ(x, y, λ) = 0
Any local maxima or minima of the function w = f(x, y, z) subject to the constraint g(x, y, z) = 0 will be among those points (x0, y0, z0) for which (x0, y0, z0 λ0) is a solution of the system:
Fx(x, y, z, λ) = 0
Fy(x, y, z, λ) = 0
Fz(x, y, z, λ) = 0
Fλ(x, y, z, λ) = 0
where F(x, y, z, λ) = f(x, y, z) + λg(x, y, z), provided that all the partial derivatives exist.