Math 201
Optimization Name:
This worksheet is designed to familiarize you with the ideas of minima and maxima, and how to
use them.
Recall that ais a critical point of f(x) if f0(a) = 0 and if ais a critical point of f(x) then f(a) is
the critical value.
Critical points are candidates for local extrema. If ais a critical point of f(x) and
•f0(x) changes from negative to positive at athen ais a local minimum.
•f0(x) changes from positive to negative at athen ais a local maximum.
When looking for global extrema, one must check either the behavior at positive and negative
infinity or the values of the function at the endpoints of the domain.
(1) Classify the critical points of the following functions.
(a) f(x) = x2−2x+ 4
(b) g(x) = x
x2+1
(c) h(x) = xln(x)
(d) k(x) = xe2x
(2) Determine the global extrema of the following functions on the specified domain.
(a) f(x) = x2−2x+ 4 on [−3,1]
(b) f(x) = x2−2x+ 4 on all real numbers
(c) g(x) = x
x2+1 on [0,∞)
(d) h(x) = xln(x) on [1
2, e]
(3) Use your knowledge of local and global extrema to solve the following problems.
(a) Farmer John is going to fence in a pasture that abuts a river. If he makes a rectangular
enclosure with 200 feet of fence, what is the maximum area he can enclose?
(b) Recall that the volume of a cylinder of height hand radius ris given by πr2h, and its
surface area is 2πr2+ 2πrh. If a can is made of alloys which cost 5 cents per square
centimeter on top and bottom but 3 cents per square centimeter on the sides, find the
cheapest way to make a can which will hold 200 cubic centimeters.
(c) Suppose advertising costs $1000 per unit (say for magazine adds), and product de-
velopment costs $20000 per unit. Suppose that the profits generated from xunits of
advertising and yunits of product development are xy2thousands of dollars. If a
company has $10000 to spend on advertising and product development together, how
should the money be allocated in order to maximize profits?
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