Lagrange Multipliers
Find the highest point on the trail, not the mountain.
Introduction
Often we want to maximize a function , but we are constrained to a specific path .
Joseph-Louis Lagrange discovered that at the maximum or minimum, the gradient of the function and the gradient of the constraint are parallel.
The Method of Lagrange Multipliers
To optimize subject to :
...and solve the system of equations. ("lambda") is the Lagrange Multiplier.
Geometric Interpretation
Interactive: Tangent Level Curves
Worked Examples
Example 1: Maximize Area
Maximize subject to .
1. Gradients:
vs .
2. Solve System:
, .
Implies . Since implies .
Max value .
Example 2: Distance to Origin
Find point on closest to origin. Maximize (dist squared).
1. Equations:
. .
(1)
(2)
(3)
2. Solve:
From (1) and (2): and .
So .
Use (3): .
Points: and .
Distance squared: 2. Distance: .
Example 3: Box Optimization
Maximize subject to (Surface Area = 12).
1. Symmetry Argument:
The equations for x, y, and z are symmetric.
This implies .
Constraint: .
Max Volume: .
Practice Quiz
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