Lagrange Multipliers Problems
Reading Assignment: 4.3
Suggested problems: 1, 3, 5, 7, 16, 18, 21, 22, 27, 30
Find the extrema of f subject to the given conditions. Please use additional sheets of paper if
room to write becomes tight.
1.
f ( x, y, z )  x  y  z; x 2  y 2  z 2  2
2.
f ( x, y)  3x  2 y; 2 x 2  3 y 2  3
3. f ( x, y, z )  x  y  z; x 2  y 2  1, 2 x  z  1
4. The ordinary UPS rate applies to packages whose length + girth is less than 108 inches, where
girth is the perimeter of the side of the box perpendicular to the length (i.e., length + girth = l +
2w + 2d where l = length, w = width, and d = depth). What are the dimensions of the largest
volume of box you can ship with regular rates?
5. Find all of the extrema of g ( x, y, z )  xyz on the surface z  e x
2
 y2
.
6. Lagrange multipliers work great for finding the minimum value of y  ax , restricted to the
curve y 2  x 2  1 for any a with 0  a  1 (what is it?) but fails quite miserably if a  1 . Use the
geometry of the situation to explain this.