Optimization Problems

Optimization Problems
The most exciting phrase to hear in science, the one that heralds the most discoveries, is not
"Eureka!" (I found it!) but "That's funny..." ~Isaac Asimov
Find the local extrema (tops of hills bottoms of valleys) of f and use the second derivative
test whenever applicable. Sketch the graph of f.
1. f ( x)  x 3  2 x 2  x  1
2. f ( x)  3x 4  4 x 3  6
3. f ( x)  8x 4  2 x 4
4. f ( x)  x 4  4 x 3  10
5. In order to redirect Shoshannah’s tears after having to part from my class after
graduation, the 12th grade math class takes a long rectangular sheet of metal, 5 inches
wide, is to be made into a rain gutter by turning up two sides so that they are
perpendicular to the sheet. How many inches should be turned up to give the tear gutter
the greatest capacity?
6. If a box with a square base and open top is to have a volume of 4cubic feet, find the
dimensions that require the least material. (Disregard the thickness of the material and the
waste in construction)
7. You need to bottle your own “swagger juice” but you decide to make a 750 ml dosage
that will keep swagger in your step for 10 days. If the container is cylindrical what is the
minimum amount of surface area that you could use to construct the bottle?
Honors/Extra Credit
1. A wire 60 inches long is to be cut into two pieces. One of the pieces will be bent into
the shape of a circle and the other into the shape of an equilateral triangle. Where should
the wire be cut so that the sum of the areas of the circle and triangle is minimized?
2. A farmer has 500 feet of fencing to enclose a rectangular field. A barn will be used as
part of one side of the field. Prove that the area of the field is greatest when the rectangle
is a square.
And here is your hint:G = M