Calculus I - Optimization
Wed, Dec 19, 2012
Name ____________________
First Derivative Test - If the sign of f (x) changes from positive to negative at a critical point x = c,
then f (x) has a relative maximum at x = c.
If the sign of f (x) changes from negative to positive at a critical point x = c,
then f (x) has a relative minimum at x = c.
 * A chart must be used in conjunction with First Derivative Test.
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Second Deriv. Test - Given f ( x) is twice differentiable on the open interval containing the CP x  c.
If f ( x)  0 on the open interval, then f (c ) is a relative minimum on the interval.
If f ( x)  0 on the open interval, then f (c ) is a relative maximum on the interval.
***CAUTION: Both of these are tests for local extrema. They cannot be used to
determine increasing/decreasing intervals or concavity.
------------------------------------------------------------------------------------------------------------------------------1. A farmer has 200 yd of fence with which to construct three sides of a rectangular pen; an existing long,
straight wall will form the fourth side. What dimesions will maximize the area of the pen? Justify.
2. A piece of sheet metal is rectangular, 5 ft wide and 8 ft long. Congruent squares are to be cut from its
four corners. The resulting piece of metal is to be folded and welded to form an open-topped box.
How should this be done to get a box of largest possible volume? Justify your answer.
3. What is the maximum possible area of a rectangle with a base that lies on the x-axis and with two
upper vertices that lie on the graph of the equation y  4  x 2 ? Justify.
4. A farmer has 600 yd of fencing with which to build a rectangular corral. Some of the fencing will be
used to construct two interval divider fences, both parallel to the same two sides of the corral. What is
the maximum possible total area for such a corral? Justify.
5. A rectangular box has a square base with edges at least 1 inch long. It has no top, and the total area of
its five sides is 300 sq in. What is the maximum possible volume of this box? Justify.
Due Thurs, Dec 20 – A.57 p.226 #2, 3, 6, 7, 9
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