Lagrange Multipliers

Optimising when you are not free to go anywhere.

12 problems · about 25 minutes · F-IF.C.7, A-CED.A.3

What this lesson teaches

The student sets up and solves constrained optimisation problems using Lagrange multipliers.

A constraint confines you to a curve rather than the whole plane.
The optimum is where the constraint curve touches a contour instead of crossing it.
That tangency means ∇f = λ∇g, which with the constraint gives three equations.

Straightforward practice. Get the method working first.

5 problems

A rectangle with perimeter 28 has maximum area. What is that area?
Answer 49
Why 49.
What is true at a constrained optimum?
Answer The constraint curve is tangent to a contour of the objective.
Why They touch rather than cross.
A rectangle with perimeter 20 has maximum area. What is that area?
Answer 25
Why A 5 by 5 square.
At the optimum, are ∇f and ∇g parallel? 1 yes, 0 no.
Answer 1
Why That is the Lagrange condition.
∇f = ⟨6, 9⟩ and ∇g = ⟨2, 3⟩ at the optimum. What is λ?
Answer 3
Why 6 divided by 2.

The same ideas with more to keep track of.

3 problems

A rectangle with perimeter 24 has maximum area. What is the side length?
Answer 6
Why A square with four equal sides.
How many unknowns are there in a two-variable Lagrange system?
Answer 3
Why x, y and one more.
Maximise x + y subject to x + y = 12. What is the maximum value?
Answer 12
Why The constraint fixes it outright.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Order the steps of a Lagrange multiplier solution.
Answer 1. Identify the objective f and the constraint g 2. Set the gradient of f equal to lambda times the gradient of g 3. Solve the system together with the constraint 4. Evaluate f at each solution and compare
Why Naming the two functions comes first.
A square of side 7 from a perimeter constraint. What is its area?
Answer 49
Why 7 squared.
The Fence: A rectangle with perimeter 36 metres has the largest area when it is a square. What is that area, in square metres?
Answer 81
Why 81 square metres.
The Equations: How many equations does a two-variable Lagrange problem produce, counting the constraint?
Answer 3
Why 3.