Spring 2006

Important: Do two of the three problems. Clearly indicate which three you want me to grade (10 points each, 20 points total, no bonus credit).

Show all work in a neat and organized fashion. Clearly indicate your answers.

1. Given:

f(x) = 7x4-28x3+41x2-26x+82,

f¢(x) = 28x3-84x2+82x-26,

f¢¢(x) = 84x2-168x+82.

Use a graphing calculator to estimate these answers to one or two decimal places.

(a) Find the critical numbers of f.

(b) Find the intervals on which f is increasing or decreasing.

(c) Find the x-coordinates of all local maxima and local minima of f.

(d) Find the intervals of concavity.

(e) Find the x-coordinates of all points of inflection.

2. If 1500 cm² of material is available to make a box with a square base and an open top, find the dimensions for the box with largest possible volume.

3. A rectangular storage container with an open top is to have a volume of 40 m³. The length of its base must be four times the width. Material for the base costs $20 per square meter. Material for the sides costs $8 per square meter. Find the dimensions for the cheapest such container.