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812 Chapter 13: Functions of Several Variables
3.10 Lagrange Multipliers
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Use Lagrange multipliers to minimize the function subject to the
following constraint:
Assume that x and y are positive.
a.
b.
c.
d.
e. no absolute minimum
____ 2. Use Lagrange multipliers to find the maximum value of where
and subject to the constraint .
a.
b.
c.
d.
e.
____ 3. Use Lagrange multipliers to maximize the function subject
to the following constraint.
Assume that x and y are positive.
a.
b.
13.10 Lagrange Multipliers 813
c.
d.
e. no absolute maximum
____ 4. Use Lagrange multipliers to minimize the function subject
to the following constraint.
Assume that x, y, and z are positive.
a.
b.
c.
d.
e.
____ 5. Use Lagrange multipliers to find the maximum value of where
and , subject to the constraint .
a.
b.
c.
d.
e.
____ 6. Use Lagrange multipliers to find the maximum value of
where subject to the constraint .
a.
maxima: ; minima:
b.
maxima: ; minima:
c.
maxima: ; minima:
814 Chapter 13: Functions of Several Variables
d.
maxima: ; minima:
e.
maxima: ; minima:
____ 7. Use Lagrange multipliers to minimize the function subject
to the following two constraints.
Assume that x, y, and z are nonnegative.
a.
b.
c.
d.
e.
____ 8. Use Lagrange multipliers to find the minimum distance from the line to
the point
a.
b.
c.
d.
e.
13.10 Lagrange Multipliers 815
____ 9. Use Lagrange multipliers to find the minimum distance from the parabola to
the point . Round your answer to two decimal places.
a.
b.
c.
d.
e.
____ 10. Use Lagrange multipliers to find the minimum distance from the circle
to the point . Round your answer to two decimal places.
a. 2.07
b.
c. 103.64
d. 10.18
e.
____ 11. Use Lagrange multipliers to find the minimum distance from the plane
to the point . Round your answer to two decimal places.
a. 3.00
b.
c.
d. 9.81
e. 96.33
____ 12. Find the highest point on the curve of intersection of the following surfaces.
Cone: , Plane:
a.
b.
c.
d.
e.
816 Chapter 13: Functions of Several Variables
____ 13. A cargo container (in the shape of a rectangular solid) must have a volume of 580
cubic feet. The bottom will cost $6 per square foot to construct and the sides and the top will cost $4
per square foot to construct. Use Lagrange multipliers to find the dimensions of the container of this
volume that has minimum cost.
a.
b.
c.
d.
e.
____ 14. Let represent the temperature at each point on the sphere
. Find the maximum temperature on the curve formed by the intersection of the
sphere and the plane .
a.
b.
c.
d.
e.
____ 15. Find the maximum production level if the total cost of labor (at
$72 per unit) and capital (at $40 per unit) is limited to $270,000, where x is the number of units of
labor and y is the number of units of capital. Round your answer to the nearest integer.
a. 42,016 units produced
b. 15,796 units produced
c. 29,115 units produced
d. 291,147 units produced
e. 19,792 units produced
13.10 Lagrange Multipliers 817
____ 16. Find the minimum cost of producing 55,000 units of a product ,
where x is the number of units of labor (at $76 per unit) and y is the number of units of capital (at $56
per unit). Round your answer to the nearest cent.
a. $68,837.29
b. $50,201.54
c. $42,169.29
d. $57,823.32
e. $54,500.66
818 Chapter 13: Functions of Several Variables
3.10 Lagrange Multipliers – Answer Section
