Multivariable Calculus Quiz: Lagrange Multiplier Interpretation

Consider the constrained optimization: minimize $f(x,y) = (x-3)^2 + (y-4)^2$ subject to $x + y = 5$. The Lagrange multiplier at the solution is $\lambda = -1$. A student argues: "Since $\lambda < 0$, increasing the constraint constant will decrease the objective function, which contradicts the fact that we're moving away from the unconstrained minimum at (3,4)." What is the flaw in this reasoning?

A company maximizes its profit function P(L,K) subject to a budgetary constraint C(L,K) = B. For the current budget B = 50000, the maximum profit is $2,100,000. Using the method of Lagrange multipliers, the value of the multiplier is found to be λ = 45. If the budget is increased to B = 51000, what is the approximate new maximum profit?

The potential energy E(x,y) of a particle is minimized while it is constrained to a path g(x,y) = c. For c = 4, the minimum energy is E_{min} = 12 joules. The corresponding Lagrange multiplier is λ = -3.5 J/m. What is the approximate minimum energy if the constraint is changed to g(x,y) = 3.8?

A factory's daily production P is measured in widgets. The cost of production C, a function of inputs x and y, is measured in dollars. To solve the problem of maximizing production for a fixed cost C = k, a Lagrange multiplier λ is used. What are the units of λ?

The maximum value of a function f(x,y) subject to a constraint g(x,y) = 100 is 750. When the constraint is changed to g(x,y) = 102, the maximum value of f becomes approximately 741. Based on this information, what is the best estimate for the Lagrange multiplier λ associated with the original problem?

A company wishes to maximize production P(x,y) subject to two separate constraints: a budget constraint B(x,y) ≤ 50000 and a resource constraint R(x,y) ≤ 300. Analyzing each constraint as an equality, the Lagrange multiplier for the budget constraint is λ_B = 1.2, and for the resource constraint is λ_R = 2.5. If the company can afford to increase the budget by $1000 or increase the resource limit by 400 units for the same cost, which is the better strategy to increase production?

A power plant minimizes its operating cost C(x,y) in dollars per hour, subject to the constraint that its power output P(x,y) is exactly k megawatts (MW). For an output of k=500 MW, the Lagrange multiplier is found to be λ = 400. Which of the following is the most accurate interpretation of this value?

A student is maximizing a utility function U(x,y) subject to a budget constraint B(x,y) = M. For a budget M = 100, they find the maximum utility is 50 utils and the Lagrange multiplier is λ = 0.8. The student is then asked to find the maximum utility for a new budget of M = 105. They calculate the new utility to be 50 + 0.8 * (5) = 54. Which statement best describes the student's result?

Let M(k) be the maximum value of a function f(x,y) subject to the constraint g(x,y)=k. The equation ∇f = λ∇g is used to find the maximum. Which expression correctly relates the Lagrange multiplier λ to the function M(k)?

When minimizing f(x,y) subject to g(x,y)=k, the Lagrange multiplier λ is positive. When maximizing the same function f(x,y) subject to the same constraint g(x,y)=k, the Lagrange multiplier μ is also positive. What can be concluded if the constraint is relaxed by increasing k to k+Δk where Δk > 0?

A function f(x,y,z) is maximized subject to two constraints: g(x,y,z) = 20 and h(x,y,z) = 30. The corresponding Lagrange multipliers are λ_1 = 8 (for g) and λ_2 = -5 (for h). What is the approximate change in the maximum value of f if the first constraint is changed to 19.5 and the second constraint is changed to 30.2?

A constrained optimization problem yields $\lambda = 12$ at the optimal solution. When the constraint constant increases by 0.1, the actual change in the objective function is 1.15, while the Lagrange multiplier predicts a change of 1.2. When the constraint constant decreases by 0.1, the actual change is -1.25. What does this pattern suggest about the relationship between the objective function and constraint?

A manufacturer wants to minimize the cost $C(x,y) = 2x^2 + 3y^2 + xy$ subject to the constraint $x + 2y = 100$, where $x$ and $y$ represent quantities of two materials. At the optimal solution, the Lagrange multiplier is $\lambda = 15$. If the constraint were changed to $x + 2y = 101$, what would be the approximate new minimum cost?

For the optimization problem: maximize $f(x,y) = xy$ subject to $x^2 + y^2 = 50$, the Lagrange multiplier at the optimal solution is $\lambda = 0.5$. If we instead optimize subject to $x^2 + y^2 = 49$, and then subject to $x^2 + y^2 = 51$, which statement about the optimal values is most accurate?

A production function $P(L,K) = 10L^{0.3}K^{0.7}$ is maximized subject to the budget constraint $20L + 50K = 1000$, where L is labor and K is capital. At the optimal allocation, $\lambda = 0.25$. The firm's economist claims that hiring one additional unit of labor (changing the constraint to $20L + 50K = 1020$) would increase production by approximately 0.25 units. What is wrong with this analysis?

A utility maximization problem has the form: maximize $U(x,y) = x^{0.6}y^{0.4}$ subject to $3x + 2y = 100$. At the optimal solution, the Lagrange multiplier is $\lambda = 0.15$. An economist interprets this as "the marginal utility of income is 0.15." If a consumer's budget increases from $100 to $105, by how much should their maximum utility change?

For the problem: maximize $f(x,y) = 2\ln(x) + 3\ln(y)$ subject to $x + 2y = 20$, the optimal solution has $\lambda = 0.4$. Two scenarios are considered: (A) changing the constraint to $x + 2y = 22$, and (B) changing the constraint to $x + 2y = 18$. The Lagrange multiplier interpretation predicts that scenario A increases the objective by 0.8 and scenario B decreases it by 0.8. Why might this approximation be less reliable for scenario B than scenario A?

A Lagrange multiplier problem has two constraints: $g_1(x,y,z) = x + y + z - 15 = 0$ and $g_2(x,y,z) = x^2 + y^2 + z^2 - 50 = 0$. At the optimal solution, $\lambda_1 = 8$ and $\lambda_2 = 1.2$. If both constraint constants change simultaneously to $x + y + z = 16$ and $x^2 + y^2 + z^2 = 48$, what is the approximate change in the optimal objective function value?

While maximizing f(x,y) subject to g(x,y)=k, the Lagrange multiplier is found to be λ=0. What is the most direct implication for the function f at the solution point?