Optimisation problem

Question

f (x,y) = 6 + x² - 2xy

s.t 3x + y = 100

Find

L''₁₁(x,y)

L''₁₂(x,y)

L''₁₁(x,y)*L''₂₂(x,y) - (L''₁₂(x,y))²

Min. f(x,y) subject to the constraint

Max. f(x,y) subject to the constraint

Answer 1

L(x,y) = 6 + x² + 2xy - λ(3x + y - 100)

L'_x = 2x + 2y - 3λ = 0; L'_y = 2x - λ = 0; L'_λ = - 3x - y + 100 = 0; 2x = λ, y = λ, x = λ/2; - 3λ/2 - λ = - 100; λ = 40

x = 20; y = 40; f(20, 40) = 6 + 400 + 1600 = 2006

If we take for example point (30, 10), then f(30, 10) = 6 + 900 + 600 = 1506 < 2006

So, at (20, 40) Max f(x, y) = f(20,40) = 2006 s.t 3x + y = 100