### Homework #1

```MATH 2177:
Mathematical Topics for Engineers
Homework #1
Due: Tuesday, September 9th (in recitation)
Show an appropriate amount of work for each problem. Write your answers neatly and
Section 1.8 – Max/Min Problems
• # 22:
Find the critical points of the following function:
f ( x, y) = x4 + y4 − 4x − 32y + 10.
Use the Second Derivative Test to determine (if possible) whether each critical point
corresponds to a local maximum, local minimum, or saddle point.
• # 33:
Repeat the directions from #22 on the function:
f ( x, y) = ye x − ey .
• # 48:
Find the absolute maximum and minimum value of the following function on given set R:
f ( x, y) = x2 + y2 − 2x − 2y;
R is the closed set bounded by the triangle with vertices (0, 0), (2, 0), and (0, 2).
• # 58:
Rectanglular boxes with a volume of 10 m3 are made of two materials. The material for the
top and bottom of the box costs \$10/m2 and the material for the sides of the box costs
\$1/m2 . What are the dimensions of the box that minimize the cost of the box?
(over)
MATH 2177 • Homework #1
Section 1.9 – Lagrange Multipliers
• # 10:
Use Lagrange multipliers to find the maximum and minimum values of f (when they exist)
subject to the given constraint:
f ( x, y) = x − y
subject to x2 + y2 − 3xy = 20.
• # 15:
Repeat the directions from #10 for:
f ( x, y, z) = x + 3y − z
subject to x2 + y2 + z2 = 4.
• # 22:
Repeat the directions from #10 for:
f ( x, y, z) = x2 + y2 − z
subject to z = 2x2 y2 + 1.
• # 26:
Find the rectangular box with a volume of 16 ft3 that has minimum surface area.
• # 31:
Find the point on the surface 4x + y − 1 = 0 closest to the point (1, 2, −3).
(No additional response required, but think about how you know this is a minimum and not
a maximum.)
```