You want to maximize f(x, y) = xy subject to the constraint x + y = 10. The Lagrange conditions give ∇f = λ∇g. Which system of equations should you solve?
Ay = λ, x = λ, x + y = 10
B2x = λ, 2y = λ, x + y = 10
Cy = 2λ, x = 2λ, x + y = 10
Dx = λy, y = λx, x + y = 10
Here f(x,y) = xy so ∇f = (y, x). The constraint is g(x,y) = x + y - 10 = 0 so ∇g = (1, 1). Setting ∇f = λ∇g gives y = λ·1 and x = λ·1, plus the constraint x + y = 10. This yields x = y = 5 and λ = 5.
Question 2 True / False
The method of Lagrange multipliers finds the global maximum or minimum of f over most of ℝ², provided the gradient condition ∇f = λ∇g is satisfied somewhere.
TTrue
FFalse
Answer: False
Lagrange multipliers only find candidates for optima *on the constraint curve* g(x,y) = 0, not over all of ℝ². Without the constraint, f might be unbounded or have a different global optimum. The method locates critical points of f restricted to the constraint; you still need to determine which candidates are maxima, minima, or neither.
Question 3 Short Answer
Geometrically, why must ∇f be parallel to ∇g at a constrained optimum?
Think about your answer, then reveal below.
Model answer: At a constrained optimum, the level curve of f must be tangent to the constraint curve g = 0. If the level curves of f crossed the constraint rather than touching it, you could move along the constraint and increase f further — so the point could not be a maximum. Tangency means the two gradients (which are perpendicular to their respective level/constraint curves) must point in the same or opposite directions, i.e., ∇f = λ∇g.
This geometric picture — level curves tangent to the constraint — is the core intuition behind the method. The scalar λ adjusts for the fact that the two gradients may have different magnitudes even when they are parallel.