Lagrange Multipliers

Optimising when you are not free to go anywhere.

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

NameDate

Figure — use these to answer the problems

These parallel contours are a linear objective. A constraint curve laid across them is optimised where it stops crossing and merely touches.

a, the x-coefficient2

b, the y-coefficient1

Figure 1

Straightforward practice. Get the method working first.

5 problems

A rectangle with perimeter 28 has maximum area. What is that area?
Answer
2
What is true at a constrained optimum?
- The constraint curve is tangent to a contour of the objective.
- The constraint curve crosses a contour at a right angle.
A rectangle with perimeter 20 has maximum area. What is that area?
Answer
At the optimum, are ∇f and ∇g parallel? 1 yes, 0 no.
Answer
∇f = ⟨6, 9⟩ and ∇g = ⟨2, 3⟩ at the optimum. What is λ?
Answer

The same ideas with more to keep track of.

3 problems

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

4 problems

9
Order the steps of a Lagrange multiplier solution.
Write 1 to 4 in the boxes to put these in order.
- Identify the objective f and the constraint g
- Set the gradient of f equal to lambda times the gradient of g
- Solve the system together with the constraint
- Evaluate f at each solution and compare
A square of side 7 from a perimeter constraint. What is its area?
Answer
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
The Equations
How many equations does a two-variable Lagrange problem produce, counting the constraint?
Answer