Lagrange Multipliers · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Best point on a fixed curve

Mathematics · Calculus & Analysis · ages 19-20
Name ______________________   Date ____________

  1. Use multipliers on f(x, y) = x + y with x squared plus y squared = 2. What is the largest value of f?

    • 2
    • 4
    • 1
    • 0

  2. Use multipliers on f(x, y) = x + y with x squared plus y squared = 2. What is the largest value of f?

    • 2
    • 4
    • 1

  3. At a constrained optimum the level curve of f just touches the constraint curve. What does that say about the two gradients?

    • Their gradients are perpendicular
    • Their gradients are parallel
    • Their gradients are both zero

  4. A rectangle has perimeter 20. What is its largest possible area?

    Answer: ______________

  5. What is the smallest value of x squared plus y squared given x + y = 4?

    Answer: ______________

  6. Wren says lambda tells how much the optimum improves when the constraint is relaxed. Is Wren right?

    Circle one:   True   False

  7. What is the smallest value of x squared plus y squared given x + y = 4?

    Answer: ______________

  8. At a constrained optimum the level curve of f just touches the constraint curve. What does that say about the two gradients?

    • Their gradients are parallel
    • Their gradients are perpendicular
    • Their gradients are both zero
    • Their gradients point in opposite fixed directions

  9. What is the smallest value of x squared plus y squared plus z squared given x + y + z = 3?

    Answer: ______________

  10. What is the smallest value of x squared plus y squared plus z squared given x + y + z = 3?

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Best point on a fixed curve W1-mt_0RPOBWZ1gs-s1

  1. 2 · Multipliers give x = y = 1, so the largest value is 1 + 1 = 2.
  2. 2 · Multipliers give x = y = 1, so the largest value is 1 + 1 = 2.
  3. Their gradients are parallel · Touching curves share a tangent line, so their normal vectors are parallel.
  4. 25 · The optimum is a 5 by 5 square, so the largest area is 25.
  5. 8 · Symmetry gives x = y = 2, so the smallest value is 4 + 4 = 8.
  6. True · Wren is right: lambda is the exchange rate at the optimum.
  7. 8 · Symmetry gives x = y = 2, so the smallest value is 4 + 4 = 8.
  8. Their gradients are parallel · Touching level curves share a tangent line, so their normal vectors, the gradients, are parallel.
  9. 3 · The optimum is x = y = z = 1, giving 1 + 1 + 1 = 3.
  10. 3 · The optimum is x = y = z = 1, giving 1 + 1 + 1 = 3.
Worksheet · LightMySky