Lagrange Multipliers: One Constraint

A constrained optimization problem requires maximizing or minimizing a quantity while satisfying a fixed condition. The rectangular pen example is constrained because the area must be maximized while the total fencing remains fixed at 100 meters. Since the barn wall provides one side, only three sides need fencing, creating a specific mathematical constraint that limits possible dimensions.

Lagrange multipliers work by setting the gradient of the objective function equal to a constant times the gradient of the constraint. At the maximum, both gradients are parallel, meaning the level curve of the area function is tangent to the constraint curve. This geometric condition, expressed as ∇A = λ∇g, ensures that the optimization respects the constraint while finding the optimal point.

Taking partial derivatives with respect to length and width yields equations that reveal the relationship between dimensions at the optimum. For the pen problem, partial derivatives show that length equals twice the width. Substituting this relationship into the fencing constraint L + 2W = 100 gives the optimal dimensions: length of 50 meters and width of 25 meters.

The Lagrange multiplier lambda accounts for the fact that the gradient of the area function and the gradient of the constraint may have different magnitudes. Lambda scales one gradient to match the other, ensuring both curves share a common tangent direction at the maximum. This constant allows the method to balance the objective function with the constraint mathematically.

The tangency condition states that the maximum occurs where a level curve of the area function touches the constraint curve. At this point, both curves have the same tangent direction, so their gradients are parallel. This geometric interpretation connects the algebraic Lagrange multiplier method to a visual understanding of how constraints limit the feasible region and determine the optimal solution.

The fencing constraint represents the total available material: one length L parallel to the wall and two widths 2W perpendicular to it sum to 100 meters. Using the Lagrange multiplier method, this constraint is combined with the area objective A = LW. Solving the resulting system shows that the optimal dimensions are L = 50 meters and W = 25 meters, maximizing the enclosed area.

Lagrange multipliers unify three perspectives: algebraically, they produce equations solved for optimal dimensions; geometrically, they identify where level curves are tangent to constraints; and physically, they respect real-world limitations like fixed fencing. This integration allows students to move between abstract mathematical concepts and practical problem-solving, understanding how optimization works across multiple representations.