Lagrange multipliers are a method for finding maximum or minimum values of a function when the inputs must satisfy a constraint. Many real problems have limits, such as fixed distance, fixed cost, fixed area, or conservation of energy. Instead of searching every allowed point, the method uses gradients to locate candidates where the best value can occur.
This makes constrained optimization systematic and easier to visualize.
Key Facts
- To optimize f(x, y) subject to g(x, y) = c, solve ∇f = λ∇g and g(x, y) = c.
- In two variables, the equations are f_x = λg_x, f_y = λg_y, and g(x, y) = c.
- In three variables, solve f_x = λg_x, f_y = λg_y, f_z = λg_z, and g(x, y, z) = c.
- The gradient ∇f points in the direction of steepest increase of f.
- At a constrained maximum or minimum, the level curve or surface of f is tangent to the constraint curve or surface.
- After solving for candidate points, evaluate f at each point to identify the maximum, minimum, or both.
Vocabulary
- Objective function
- The function f that you want to maximize or minimize.
- Constraint
- An equation such as g(x, y) = c that limits the points allowed in the problem.
- Gradient
- The vector of partial derivatives that points in the direction of greatest increase of a function.
- Lagrange multiplier
- The scalar λ that relates the gradient of the objective function to the gradient of the constraint.
- Level curve
- A curve made of points where a function has the same constant value.
Common Mistakes to Avoid
- Forgetting the constraint equation is wrong because ∇f = λ∇g alone does not guarantee that the point is allowed.
- Treating λ as the answer is wrong because λ is an auxiliary variable, while the actual optimum is found from the point and the value of f there.
- Assuming every solution is a maximum is wrong because Lagrange multipliers give candidate points that must be tested and compared.
- Using the method when ∇g = 0 without checking is wrong because the standard gradient condition may fail at singular points of the constraint.
Practice Questions
- 1 Find the maximum and minimum of f(x, y) = xy subject to x^2 + y^2 = 8.
- 2 Use Lagrange multipliers to minimize f(x, y, z) = x^2 + y^2 + z^2 subject to x + 2y + 2z = 9.
- 3 Explain why the gradient of the objective function must be parallel to the gradient of the constraint at a smooth constrained maximum or minimum.