To minimize f(x, y) subject to the constraint g(x, y) = 0, the Lagrange condition requires:
A∇f = 0 at the solution
B∇f = λ∇g for some scalar λ
C∇g = 0 at the solution
Df(x, y) = g(x, y) at the solution
The Lagrange condition is ∇f = λ∇g: the gradient of the objective must be parallel to the gradient of the constraint at any constrained optimum. The gradient of f is not zero in general (that would be an unconstrained critical point), and the constraint gradient ∇g points normal to the constraint surface.
Question 2 True / False
At a constrained optimum, the gradient of the objective function f should equal the zero vector.
TTrue
FFalse
Answer: False
At a free (unconstrained) optimum, ∇f = 0. But at a constrained optimum, ∇f is typically nonzero — it is parallel to ∇g (the constraint normal). The condition ∇f = λ∇g says the objective 'wants' to move in the same direction the constraint prevents it from moving. Setting ∇f = 0 would mean the unconstrained optimum already satisfies the constraint, which is a coincidence, not the general case.
Question 3 Short Answer
A factory maximizes output f(x, y) subject to a budget constraint g(x, y) = 5000. The Lagrange multiplier λ = 12. What does this value mean practically?
Think about your answer, then reveal below.
Model answer: Each additional unit of budget (e.g., one extra dollar) would allow approximately 12 more units of output. λ is the marginal value of the constraint.
The Lagrange multiplier measures the sensitivity of the optimal objective value to the constraint bound. If the budget increased from 5000 to 5001, the maximum output would increase by approximately λ = 12. In economics this is called the shadow price of the constraint — how much you would be willing to pay for one more unit of the scarce resource.