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The following is a graph of the
first derivative f’(x)
of a function y = f(x).
You may assume f(x) is defined for all real numbers.
y
Use this graph of
f’(x)
to answer the
2
y = f’(x)
following questions about the graph of
f(x).
1
76543–21
1 2 3
4
5
6
7 8
x
2
a. On what interval(s) is the graph of f(x) concave up?
ANSWER
: The graph of f(x) is concave up on the intervals in which f’’(x) > 0, ie on the
intervals in which f’(x) is increasing, that is (
∞
, 1) and (4,+
∞
).
b. On what interval(s) is the graph of f(x) concave down?
ANSWER:
The graph of f(x) is concave down on the intervals in which f’’(x) < 0, ie on the
intervals in which f’(x) is decreasing, that is (1,4).
c. Give the x coordinate of the point(s) of inflection.
ANSWER:
The points of inflection are the points about which the graph changes concavity.
These may be the points at which f’’(x) DNE or at which f’’(x) = 0. The x coordinates for
the points of inflection for the graph of y = f(x) are x = 1 and x = 4. (In this example
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This note was uploaded on 09/25/2011 for the course CALCULUS 135 taught by Professor Augustarainsford during the Spring '11 term at Rutgers.
 Spring '11
 Augustarainsford
 Derivative

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