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📝 Lagrange multipliers with three variables (14 MCQs)

📖 From Calculus • 14. Partial Derivatives Calculus • 14 questions available

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What is Lagrange multipliers with three variables?

Definition:
Extends to $f(x,y,z)$ with constraint $g(x,y,z)=k$ via $\nabla f = \lambda \nabla g$ in $\mathbb{R}^3$ ; or two constraints $\nabla f = \lambda \nabla g + \mu \nabla h$ .

Example:
Maximize $f=xyz$ on sphere $x^2+y^2+z^2=1$ : symmetry gives $x=y=z=1/\sqrt{3}$ , max $1/(3\sqrt{3})$ .

Reason:
Three-variable version handles volumetric constraints and multi-constraint systems common in physics, engineering design, and resource allocation.

Easy

Medium

Hard

📝 All Lagrange multipliers with three variables MCQs

Q1. A manufacturing firm models output as $f(x,y,z)$ subject to budget constraint $g(x,y,z)=k$ . At optimum, $\nabla f = \lambda \nabla g$ . If $\lambda$ increases when $k$ rises, what does this imply about marginal productivity of capital?

A.Marginal productivity is diminishing

B.Marginal productivity is increasing ✅

C.The constraint is non-binding

D.Output elasticity equals zero

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: When the Lagrange multiplier lambda increases with a relaxed constraint, it indicates that each additional unit of the constrained resource yields greater marginal gain in the objective function. This reflects increasing marginal productivity relative to the constraint, contrary to typical diminishing returns assumptions in standard optimization scenarios.

Q2. In optimizing $f(x,y,z)$ under $g(x,y,z)=c$ , a student sets up $\mathcal{L}=f-\lambda g$ but solves $\nabla f = \nabla g$ without $\lambda$ . Which error best describes the flaw?

A.Incorrect partial derivatives

B.Misapplication of chain rule

C.Omission of proportionality constant ✅

D.Wrong sign convention

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The Lagrange multiplier method requires that gradients be parallel, not necessarily equal. Omitting lambda assumes unit proportionality between gradient vectors, which only holds coincidentally. The multiplier serves as the scaling factor ensuring directional alignment while allowing magnitude differences essential for satisfying the constraint equation properly.

Q3. Given contour surfaces of $f$ and constraint surface $g=c$ intersect tangentially at point P. If moving along $g=c$ near P causes $f$ to decrease in all directions, what can be concluded?

A.P is a saddle point

B.P is a local maximum ✅

C.P is a local minimum

D.Insufficient information

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Tangential intersection indicates a critical point where gradients are parallel. When the objective function decreases in every feasible direction along the constraint manifold, the point represents a local maximum. This geometric interpretation confirms optimality without requiring second-derivative tests or bordered Hessian calculations for verification.

Q4. An economist maximizes utility $U(x,y,z)$ subject to $px+qy+rz=m$ . After solving, they find $\lambda=0.5$ . If income $m$ increases by $10, approximate change in maximum utility is:

A.Exactly 5 units

B.At most 5 units

C.Approximately 5 units ✅

D.Cannot determine from lambda alone

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The Lagrange multiplier represents the rate of change of the optimal value with respect to the constraint constant. For small changes, delta U approximately equals lambda times delta m. This linear approximation holds locally but may deviate for larger perturbations due to curvature effects in the utility surface.

Q5. Consider maximizing $xyz$ subject to $x+y+z=12$ with $x,y,z>0$ . A student claims symmetry implies $x=y=z=4$ is always optimal regardless of constraint form. What is wrong with this reasoning?

A.Symmetry only applies to linear constraints

B.Product functions lack symmetric optima

C.Constraint must also be symmetric in variables ✅

D.Boundary conditions were ignored

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Symmetric solutions arise only when both objective and constraint exhibit identical permutation symmetry across variables. If the constraint breaks this symmetry through unequal coefficients or nonlinear terms, the optimum shifts away from equal values. Students often overgeneralize symmetry arguments without verifying structural compatibility between function and constraint geometries.

Q6. Graph shows level surfaces of $f$ nested concentrically and constraint surface cutting through them. At intersection point Q, level surface appears flattened against constraint. What does this visual feature indicate about the bordered Hessian determinant?

A.Determinant is positive

B.Determinant is negative

C.Determinant is zero ✅

D.Sign cannot be inferred visually

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Flattened contact between level and constraint surfaces suggests degenerate critical behavior where second-order sufficient conditions fail. This corresponds to a zero bordered Hessian determinant, indicating possible inflection or higher-order tangency rather than strict extremum. Visual diagnosis complements analytical tests when algebraic verification becomes computationally intensive or ambiguous.

Q7. Engineer minimizes material cost $C(x,y,z)$ for fixed volume $V(x,y,z)=V_0$ . Solution yields $\lambda<0$ . Student argues negative multiplier violates economic intuition since costs should increase with volume. How should this be interpreted?

A.Calculation error occurred

B.Minimization reverses multiplier sign ✅

C.Volume constraint was formulated incorrectly

D.Negative lambda is physically impossible

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: In minimization problems, the Lagrange multiplier sign depends on formulation convention. When minimizing cost subject to equality constraint, negative lambda correctly indicates that relaxing the volume requirement reduces minimum cost. The magnitude still represents marginal cost sensitivity; only the sign reflects optimization direction relative to constraint orientation.

Q8. Compare two approaches for optimizing $f(x,y,z)$ under $g(x,y,z)=c$ : Method A uses substitution eliminating z; Method B uses Lagrange multipliers. Under which condition does Method A risk losing valid solutions?

A.When g is nonlinear in z

B.When partial g/partial z vanishes somewhere ✅

C.When f has discontinuities

D.When constraint defines multiple branches

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Substitution requires solving g= c explicitly for z as a function of x and y. Where the implicit function theorem fails because partial derivative vanishes, the elimination approach cannot capture points with vertical tangent planes in constraint surface. Lagrange multipliers handle such singularities naturally since they do not require explicit variable isolation or functional representation.

Q9. Biologist models population fitness $F(a,b,c)$ under resource constraint $R(a,b,c)=R_0$ . Optimal allocation satisfies $\nabla F=\lambda\nabla R$ . Field data shows actual populations deviate systematically from predicted optimum. Which explanation best reconciles theory with observation?

A.Model parameters are miscalibrated

B.Evolutionary dynamics prevent instantaneous optimization

C.Constraint function misrepresents true limitations

D.Fitness landscape contains unmodeled variables ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: Theoretical optima assume perfect adaptation and complete knowledge of fitness landscape. Systematic deviations suggest missing ecological factors like predation pressure, temporal variability, or genetic constraints not captured in three-variable model. Rather than rejecting optimization framework, researchers should expand state space or incorporate stochastic elements reflecting biological reality beyond simplified deterministic formulations.

Q10. Student computes bordered Hessian for constrained optimization and obtains alternating signs starting with positive. They conclude local minimum exists. However, verification reveals saddle point. What likely caused misclassification?

A.Incorrect border row ordering

B.Miscalculated second partials

C.Confused sign pattern for n=3 ✅

D.Used wrong number of constraints

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: For three variables with one constraint, sufficient condition for local minimum requires bordered Hessian determinant to have sign opposite to (-1)^m where m equals number of constraints. With m=1, determinant should be negative for minimum. Alternating signs starting positive actually indicate maximum or saddle depending on dimension. Sign conventions vary across textbooks causing frequent classification errors.

Q11. Physics problem maximizes entropy $S(E,V,N)$ subject to energy conservation $E=E_0$ . Result gives temperature via $1/T=\partial S/\partial E$ . If instead we constrain volume $V=V_0$ , the multiplier relates to pressure. Why do different constraints yield different physical quantities?

A.Entropy definition changes with constraint

B.Multipliers encode conjugate variables specific to constrained quantity ✅

C.Thermodynamic potentials are interchangeable

D.Constraint choice alters system boundaries

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Lagrange multipliers in thermodynamics represent intensive variables conjugate to extensive constrained quantities. Energy constraint yields inverse temperature because entropy derivative with respect to energy defines thermal equilibrium. Volume constraint produces pressure-over-temperature ratio since mechanical work couples to volume changes. Each multiplier encodes the thermodynamic force balancing the imposed restriction within fundamental relation structure.

Q12. Optimization of $f=x^2+y^2+z^2$ subject to $xyz=8$ yields candidate points. Second-order test is inconclusive at symmetric solution. Without higher derivatives, how can one verify global minimum status?

A.Function is coercive on constraint set ✅

B.Constraint defines compact manifold

C.Objective grows unbounded away from origin

D.All candidates satisfy first-order conditions

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: While compactness guarantees existence of global extrema, verifying a particular critical point requires analyzing objective behavior on entire feasible set. Since squared norm tends to infinity as any coordinate approaches zero or infinity while maintaining product constraint, the function is coercive on the constraint manifold. Combined with unique critical point in positive orthant, this establishes global minimality despite inconclusive second-order test.

Q13. Economic model maximizes profit $\pi(K,L,M)$ subject to production quota $Q(K,L,M)=Q_0$ . Manager observes that doubling all inputs doubles optimal profit. What property must hold for the Lagrange multiplier?

A.Lambda scales inversely with input levels

B.Lambda remains invariant under proportional scaling ✅

C.Lambda doubles with output target

D.Lambda depends only on input ratios

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Homogeneity of degree one in production function implies Euler's theorem applies. When profit function inherits this homogeneity through linear pricing, optimal multiplier stays constant under proportional scaling of targets and inputs. This scale invariance reflects constant returns to technology where marginal value of quota relaxation does not depend on absolute production level, only on technical coefficients and market prices.

Q14. Researcher optimizes drug efficacy $E(d,t,p)$ under toxicity constraint $T(d,t,p)=T_{max}$ . Clinical trial data shows optimal dose-response curve bends unexpectedly near boundary. Standard Lagrange analysis predicts smooth tangency. What modification addresses this discrepancy?

A.Add inequality constraints with KKT conditions ✅

B.Reparameterize using logarithmic transforms

C.Include interaction terms in efficacy model

D.Apply penalty function methods

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Unexpected bending near constraint boundary suggests active inequality restrictions beyond simple equality assumption. Karush-Kuhn-Tucker framework extends Lagrange method to handle binding and non-binding inequality constraints simultaneously. Complementary slackness conditions allow identification of regions where toxicity limit binds versus interior optima, capturing piecewise behavior that pure equality-constrained analysis misses when physiological thresholds create kinks in feasible response surface.

🔗 Related Topics (MCQs)

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