1. Multiple Choice The number $\lambda$ in the equations ${\nabla }f(x,y)= \lambda {\nabla }g(x,y)$ is called a [(a) factor, (b) Lagrangian, (c) Lagrange multiplier].

2. True or False Extreme values of the function $z=f( x,y)$ subject to the constraint $g( x,y) =0$ are found as solutions of the system of equations ${\nabla }f(x,y)=\lambda {\nabla }g(x,y)$ and $g(x,y)=0.$

3. True or False Lagrange multipliers can be used only for functions of two variables.

4. True or False When using Lagrange multipliers to find the extreme values of a function, the number of Lagrange multipliers introduced depends on the number of variables in the function.

5. $f(x,y)=3x+y$, $g(x,y)=xy-8=0$

6. $f(x,y)=3x+y+4$, $g(x,y)=xy-1=0$

7. $f(x,y)=3x+y+4$, $g(x,y)=x^{2}+4y^{2}-1=0$

8. $f(x,y)=3x+y$, $g(x,y)=x^{2}+y^{2}-4=0$

9. $f(x,y)=x-2y^{2}$, $g(x,y)=x^{2}+y^{2}-1=0$

10. $f(x,y)=x^{2}+4y^{3}$, $g(x,y)=x^{2}+2y^{2}-2=0$

11. $f(x,y)=2xy$, $g(x,y)=x^{2}+y^{2}-2=0$

12. $f(x,y)=xy$, $g(x,y)=9x^{2}+4y^{2}-36=0$

13. $f(x,y)=x^{2}-4xy+4y^{2}$, $g(x,y)=x^{2}+y^{2}-4=0$

14. $f(x,y)=9x^{2}-6xy+y^{2}$, $g(x,y)=x^{2}+y^{2}-25=0$

15. $f(x,y,z)=4x-3y+2z$, $g(x,y,z)=x^{2}+y^{2}-6z=0$

16. $f(x,y,z)=x^{2}+2y^{2}+z^{2}$, $g(x,y,z)=2x-3y+z-6=0$

17. Find the absolute maximum and the absolute minimum of the function $f(x,y)=x^{2}+y^{2}+4xy$ subject to the constraint $x^{2}+y^{2}\leq 2$.

18. Find the absolute maximum and the absolute minimum of the function $f(x,y)=2x^{2}+y^{2}$ subject to the constraint $x^{2}+y^{2}\leq 4$.

19. Find the absolute maximum and the absolute minimum values of the function $f(x,y,z)=x^{2}+y^{2}+z^{2}$ subject to the constraints $z^{2}=x^{2}+y^{2}$ and $x+y-z+1=0$.

20. Find the absolute minimum value of $w=x^{2}+y^{2}+z^{2}$ subject to the constraints $2x+y+2z=9$ and $5x+5y+7z=29$.

21. Minimizing Distance Find the point on the line $x-3y=6$ that is closest to the origin.

22. Minimizing Distance Find the point on the plane $2x+y-3z=6$ that is closest to the origin.

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23. Find the maximum product of two numbers $x$ and $y$ subject to $x+2y=21.$

24. Find the minimum quotient of two positive numbers $x$ and $y$ subject to $4x+2y=100.$

25. 1. (a) Find the points on the intersection of $4x^{2}+y^{2}+z^{2}=16$ and the plane $3-y=0$ that are farthest from the origin.
    2. (b) Find the points that are closest to the origin.

26. Find the point on the intersection of the sphere $x^{2}+2x+y^{2}+ z^{2}=16$ and the plane $3x+y-z=0$ that is farthest from the origin. Also find the point that is closest to the origin.

27. At which points on the ellipse $x^{2}+2y^{2}=2$ is the product xy a maximum?

28. Maximizing Volume Find the dimensions of an open-topped box that maximizes volume when the surface area is fixed at 48 square centimeters.

29. Manufacturing and Design Find the optimal dimensions for a can in the shape of a right circular cylinder of fixed volume $V$. That is, find the height $h$ and the radius $r$ of the can in terms of $V$ so that the surface area is minimized. Assume the can is closed at the top and at the bottom.

    Source: Problem submitted by the students at Minnesota State University

30. Minimizing Materials A manufacturer receives an order to build a closed rectangular container with a volume of 216 m $^3$. What dimensions will minimize the amount of material needed to produce the container?

31. Cost of a Box An open-topped box has a volume of 12 m $^3$ and is to be made from material costing $1 per square meter. What dimensions minimize the cost?

32. Cost of a Box A rectangular box is to have a bottom made from material costing $2 per square meter, while the top and sides are made from material costing $1 per square meter. If the volume of the box is to be 18 m $^3$, what dimensions will minimize the cost of production?

33. Carry-On Luggage Requirements The linear measurements (length + width + height) for luggage carried onto a Delta Airlines plane must not exceed $45 \rm{inches}$. Find the dimensions of the rectangular suitcase of greatest volume that meets this requirement.

    

    Source: Delta Airlines, 2012.

34. Extreme Temperature Suppose that $T=T(x,y,z)= 100x^{2}yz$ is the temperature (in degrees Celsius) at any point $(x,y,z)$ on the sphere given by $x^{2}+y^{2}+z^{2}=1$. Find the points on the sphere where the temperature is greatest and least. What is the temperature at these points?

35. Fencing A farmer has 340 m of fencing for enclosing two separate fields, one of which is to be a rectangle twice as long as it is wide and the other a square. The square field must enclose at least 100 square meters (m$^2$), and the rectangular one must enclose at least 800 m$^2$.

    1. (a) If $x$ is the width of the rectangular field, what are the maximum and minimum values of $x$?
    2. (b) What is the greatest number of square meters that can be enclosed in the two fields?

36. Fencing in an Area A Vinyl Fence Co. prices its Cape Cod Concave fence, which is 3 ft tall, at $21.53 per linear foot. A home builder has $5000 available to spend on enclosing a rectangular garden. What is the largest area that can be enclosed?

    Source: A Vinyl Fence Co. San Jose, California, 2012.

37. Joint Cost Function Let $x$ and $y$ be the number of units (in thousands) of two products manufactured at a factory, and let $C=18x^{2}+9y^{2}$ in thousands of dollars be the joint cost of production of the products. If $x+y=5400$, find $x$ and $y$ that minimize production cost.

38. Production Function The production function of a company is $P( x,y) =x^{2}+3xy-6x,$ where $x$ and $y$ represent two different types of input. Find the amounts of $x$ and $y$ that maximize production if $x+y=40.$

39. Economics: The Cobb–Douglas Model Use the Cobb–Douglas production model $P=1.01K^{0.25} L^{0.75}$ as follows: Suppose that each unit of capital $\left( K\right)$ has a value of $\$175$ and each unit of labor $\left( L\right)$ has a value of $125.

    1. (a) If there is a total of $\$175{,}000$ to invest in the economy, use Lagrange multipliers to find the units of capital and the units of labor that maximize the total production in the manufacturing sector of the economy.
    2. (b) What are the maximum units of production that the manufacturing sector of the economy could generate under these conditions?

40. The surface $xyz=-1$ is cut by the plane $x+y+z=1$, resulting in a curve $C$. Find the points on $C$ that are nearest to the origin and farthest from the origin.

41. Find the minimum value of $w=x^{2}+y^{2}+z^{2}$ subject to the constraints $2x+y+2z=9$ and $5x+5y+7z=29$.

42. Maximizing Volume A closed rectangular box of fixed surface area and maximum volume is a cube. Use Lagrange multipliers to confirm this fact.

43. Maximizing Volume A closed cylindrical can of fixed surface area and maximum volume has a height equal to the diameter of its base. Use Lagrange multipliers to confirm this fact.

44. Find the points of intersection of the plane $x+y+z=1$ and the hyperboloid $x^{2}+y^{2}-z^{2}=1$ nearest the origin.

45. At what points on the union of the two curves $x^{2}+y^{2}=1$ and $x^{3}+y^{3}=1$ is the function $f(x,y)=x^{4}+y^{4}+4$ a maximum? At what points is it a minimum?

46. Find the extreme values of $f(x,y,z)=xyz$ on the surface $x^{2}+y^{2}+z^{2}=1$.

47. Minimize $x^{4}+y^{4}+z^{4}$ subject to the constraint $Ax+By+Cz=D$.

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48. Use Lagrange multipliers to show that the triangle of largest perimeter that can be inscribed in a circle of radius $R$ is an equilateral triangle.

49. Find the point of the paraboloid $z=2-x^{2}-y^{2}$ that is closest to the point $(1,1,2)$.

50. What points of the surface $xy-z^{2}-6y+36=0$ are closest to the origin?