ST 552

Calculus on a Computer

Assignment 7: Max – Min Problems

1. Find the critical points of on the interval [0, 5]. Identify the type of each. Plot.

2. Find the local and global minima and maxima of on [-1, 2]. Plot.

3. If two numbers add to 12, what is the largest their product can be? How about the sum of their cubes?

4. A poster is to have 150 square inches of printed area. If we want a 2 inch margin at the top, and 1 inch margins at the bottom and sides, what dimensions use the smallest amount of paper?

5. The soda can is to hold 355 ml. If the material for the top of the can costs three times as much per square centimeter as the material for the bottom and sides, what is the most economic size?

6. An off-shore oil rig is 3 km away from the shore. We wish to pump the oil to a location 10 km along the shore away from the rig. If it costs 1.8 times as much per kilometer to build an underwater pipeline as one on land, how should we design our pipeline?

7. To make pompoms in our school colors, we will have expenses of $100 to rent the Acme PomPomPlant, and then $0.25/ pompom in materials. We believe that we can sell 500 pompoms if we charge $1, but we will only sell 300 if we charge $1.50. Assume that the number sold is a linear function of price. How much should we charge to maximize our profit?

8. The strength of a rectangular beam is proportional to. What are the dimensions of the strongest rectangular beam that can be cut out of a 12 inch diameter log?

9. A line in the first quadrant is tangent to the graph of. How can we minimize the area between the line and the axes?

10. Consider the function. For this function, is never (check this). Must the maxima and minima of the function on the interval [-1, 2] be at the endpoints? Explain.