Math 2413
Notes 3.5
Section 3.5 – Concavity and Points of Inflection
Definition: Let f(x) be a differentiable on an interval I. The graph of f is:
i.
Concave up on an open interval I if f’ is increasing on I.
ii.
Concave down on an open interval I if f’ is decreasing on I.
Points of Inflection.
Definition: A point where the graph of a function has a tangent line and where the concavity changes is a point
of inflection.
Examples: For each graph, find the intervals for which the graph is concave up and concave down. Also, find
all points of inflection.
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Math 2413
Notes 3.5
Theorem: The Second Derivative Test for Concavity
Let f(x) be a twice – differentiable on an interval I . The graph of f is:
i.
Concave up on an open interval I if f " ( x)  0 .
ii.
Concave down on an open interval I if f " ( x )  0 .
Fact: At the points of inflection c, f (c)  , either f " (c)  0 or f " (c ) fails to exist.
Example 1: Given function f ( x )  x 3  3 x  3 . Determine the intervals for which the graph of the function is
concave up, concave down and find all points of inflection if any.
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Math 2413
Notes 3.4
4
2
Example 2: Given function f ( x)  x  2 x . Determine the intervals for which the graph of the function is
concave up, concave down and find all points of inflection if any.
Example 3: Given function f ( x)  x  sin x, 0  x  2 . Determine the intervals for which the graph of the
function is concave up, concave down and find all points of inflection if any.
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Math 2413
Notes 3.4
Example 4: Given function f ( x)  x 8  x 2 . Determine the intervals for which the graph of the function is
concave up, concave down and find all points of inflection if any.
Example 5: Given function f ( x) 
8x
. Determine the intervals for which the graph of the function is
x 4
2
concave up, concave down and find all points of inflection if any.
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Math 2413
Notes 3.4
Example 6: Given function f ( x)  2 x 3  6 x 2  3 . Use the first and second derivative tests to determine when the
function is increasing, decreasing, concave up, and concave down. Then f ind all local extrema and inflection points.
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Math 2413
Notes 3.4
Example 7: Given function f ( x)  x 5  5 x 4 . Use the first and second derivative tests to determine when the function
is increasing, decreasing, concave up, and concave down. Then find all local extrema and inflection points.
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Math 2413
Notes 3.4
Example 8: Given function f ( x)  sin x cos x, on 0  x  2 . Use the first and second derivative tests to determine
when the function is increasing, decreasing, concave up, and concave down. Then find all local extrema and inflection
points.
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Math 2413
Example 9: Given function f ( x) 
Notes 3.4
x2  3
. Use the first and second derivative tests to determine when the function is
x2
increasing, decreasing, concave up, and concave down. Then find all local extrema and inflection points.
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Math 2413
Notes 3.4
2
Example 10: Given function f ( x)  x 5 . Use the first and second derivative tests to determine when the function is
increasing, decreasing, concave up, and concave down. Then find all local extrema and inflection points.
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Math 2413
Notes 3.4
Example 11: The graph of f′(x) is shown below. Give the interval(s) where the graph of f(x) is concave down.
Example 12: The graph of f′(x) is shown below. Give the interval(s) where the graph of f(x) is concave down.
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