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Chapter 7.4 Optimization I
______________ Maximum (y-value): Given a function
f defined on its domain D f, iff (x) f (c)for all x in
D , then f (c)is the ______________ maximum value.
f
______________ Minimum (y-value): Given a function
f defined on its domain D f, iff (x) f (c)for all x in
D f, then f (c)is the ______________ minimum value.
Extreme Value Theorem: Existence Theorem A
____________ function f defined on some closed
interval [a,b] will have both an ______________
maximum and an ______________ minimum on the
interval.
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p. 386 2
Examples
Q1: Find the absolute extrema, if they exist, for the
following functions defined on the given interval.
3 2
a) f (x) x 6x 1 , 1,5]
b) g(x) 3 x 2 , [1,4] 3
2 2
c) h(x) (x 1) (x 1) ,[ 2,3]
3 3
d) k(x) 5x x ,[ 1,4] 4
Q2: Indicate all absolute and relative extrema on the graph of
the function f below for the interval [ 8,6] .
Q3: Indicate all absolute and relative extrema on the graph of
the function f below for the interval [ 1,3] . 5
Q4: The demand equation for a manufacturer’s product is
80 x
p 4 with 0 x 80 , where x is the number of units and
p is price per unit in dollars.
a) At what value of x will there be maximum revenue?
b) What is the maximum revenue? 6
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Q5: A bicycle manufacturer has costs C(x) 60x 15x 10
and demands p 90 x for 0 x 90 bicycles per week,
where p is the price in dollars per bicycle.
a) Find the maximum weekly revenue.
b) Find the maximum weekly profit. 7
c) Show that marginal revenue and marginal cost are the
same at the point where the maximum profit occurs.
d) Find the price the manufacturer should charge to realize
maximum profit. 8
Chapter 7.5 Optimization II
Optimization Solving Guidelines: I suggest you do the
following steps in ______________ in order to solve an
optimization problem posed on a function f .
1. Read the problem at least _________.
2. If possible, ___________ a diagram and ___________
pertinent parts.
3. _

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