Math 161, Fall 2003
Exam 3
Name:
Answer all questions.
1. Prove from the definitions that if f0(a)>0 then fdoes not have a local maxi-
mum at a. (15 Points)
2. Prove that if f0(x) = g0(x) on (a, b) then f(x)−g(x) is constant on (a, b).
(You may quote, rather than proving, the relevant corollary to the Mean Value
Theorem from class.) (10 Points)
3. Find the maximum and minimum of
g(x) = x2− |x|
on [−1,2] (15 Points)
4. You are given a circular piece of plastic with radius 10 cm. How do you make
a conical drinking cup from it with maximum volume? (15 Points)
5. Find the function to be minimized and the interval on which to minimize it for
the following situation: What are the proportions of the pyramid with a square
base with a given volume which lead to least total surface area? (10 Points)
6. Where is f(x) = (x−2)(x−4)
x2−1increasing? (10 Points)
7. Where is h(x) = x3
x2+ 1 concave up? (10 Points)
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