Name: lagrange multipliers on an intersection curve
Uploaded: 2026-06-08T09:38:54.555Z
Duration: 1 min 28 s
Channel: JoVE

Consider a rectangular shipping carton with dimensions x, y, and z. The goal is to minimize the surface area to reduce material costs. At the same time, the carton must meet two conditions: a fixed volume and a shipping size requirement.In this problem, surface area is the quantity being minimized, while volume and shipping size are the constraints.Geometrically, each constraint forms a surface in three-dimensional space. Any carton design that meets both constraints must lie on the curve where these two surfaces intersect.To find the optimal point on this curve, the method of Lagrange multipliers with two constraints is used. This introduces two multipliers, lambda and mu. At this point, the gradient of the surface area is a linear combination of the gradients of the two constraints. This means the gradient of the surface area lies in the plane spanned by the two constraint gradients.Next, partial derivatives with respect to each variable are computed. These derivative equations, together with the two constraint equations, form a system. Solving this system gives the values of x, y, and z that minimize the surface area while meeting both requirements.

lagrange multipliers: two constraints

The method of Lagrange multipliers with two constraints is used to optimize a function subject to two independent constraints. In many applications, the objective function represents a quantity to be maximized or minimized, such as cost, area, distance, or energy. The two constraints represent requirements that the solution must satisfy, such as fixed volume, limited resources, or prescribed dimensions.For a function of three variables, each constraint forms a surface in three-dimensional space. Feasible points are those that satisfy both constraints simultaneously. These points usually lie along the curve where the two constraint surfaces intersect. Since optimization is limited to this curve, the goal is not to find the maximum or minimum over all space, but only among the points that meet both conditions.At an optimal point, the objective function cannot increase or decrease along the feasible curve. The gradient of the objective function is therefore perpendicular to the tangent direction of the curve. The gradients of the two constraint functions are also perpendicular to their respective constraint surfaces. Together, these two constraint gradients define a plane of normal directions. For this reason, the gradient of the objective function can be expressed as a linear combination of the two constraint gradients.To apply the method, two multipliers, commonly denoted by lambda and mu, are introduced. Partial derivatives are then taken with respect to each variable. These derivative conditions are combined with the two original constraint equations to form a complete system. Solving this system gives the candidate points for the constrained optimum. These points are then checked in the context of the problem to identify the required maximum or minimum.

Lagrange Multipliers: Two Constraints

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Education

JoVE Core

Calculus

Unit

Partial Derivatives and Gradients

KEY TERMS AND CONCEPTS

lagrange multipliers on an intersection curve

Lagrange multipliers with two constraints help find a maximum or minimum when a function must satisfy two conditions at once. The goal may be to optimize cost, area, distance, or energy, but only among solutions that meet both requirements. In many problems, the constraints represent fixed volume, limited resources, or prescribed dimensions.

For a function of three variables, each constraint makes a surface in three-dimensional space. A feasible point is a point that satisfies both constraints. These points usually lie on the curve where the two constraint surfaces intersect. Because the search is restricted to that curve, the answer is not the best point in all space. It is only the best point among the points that meet both conditions.

At an optimal point, the objective function cannot increase or decrease along the feasible curve. That means the gradient of the objective function is perpendicular to the curve's tangent direction. The gradients of the two constraint functions are also perpendicular to their own surfaces. Together, those two constraint gradients define a plane of normal directions. So the gradient of the objective function can be written as a linear combination of the two constraint gradients.

To solve the problem, two multipliers are introduced, usually called lambda and mu. Partial derivatives are taken with respect to each variable, and the derivative conditions are combined with the two original constraint equations. This creates a complete system of equations. Solving that system gives the candidate points for the constrained optimum. Those points are then checked in the context of the problem to find the required maximum or minimum.

Lagrange Multipliers on an Intersection Curve

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Lagrange multipliers with two constraints help find a maximum or minimum when a function must satisfy two conditions at once. The goal may be to optimize cost, area, distance, or energy, but only among solutions that meet both requirements. In many problems, the constraints represent fixed volume, limited resources, or prescribed dimensions. For a function of three variables, each constraint makes a surface in three-dimensional space. A feasible point is a point that satisfies both constraints.

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key terms and concepts

Learn how two constraints limit optimization to an intersection curve, and how lambda and mu help solve for candidate maximum or minimum points.

Lagrange Multipliers on an Intersection Curve | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Lagrange Multipliers on an Intersection Curve

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