3.4 Concavity and the Second Derivative Test
The second derivative is useful for determining:
• the concavity of a function,
• the location of possible points of inflection,
• or whether a critical number on the graph is indeed a relative maximum or relative minimum.
Test for Concavity
Let f be a function whose second derivative exists on an open interval I.
1. If f ’’(x) > 0 for all x in I, then the graph is concave up in I.
2. If f ’’(x) < 0 for all x in I, then the graph is concave down in I.
To determine the concavity for a function, locate the x-values for which f ’’(x) = 0 or f ’’(x) is undefined. Test
the intervals between these values.
Points of Inflection
A possible point of inflection on a graph can be determined by finding the x values for which the f ’’(x) = 0 or
is undefined.
Test for Relative Maximum or Minimum
Let f be a function such that f ’(c) = 0 and the second derivative exists on an open interval containing c.
1. If f ’’(c) > 0, then f(c) is a relative minimum (function is concave up at that point).
2. If f ’’(c) < 0, then f(c) is a relative maximum (function is concave down at that point).
3. If f ’’(c) = 0, then the test fails, you must use the First Derivative Test.
Example 1: Determine the points of inflection and discuss the concavity of the graph: f ( x ) = 4 x 3 ! 12 x
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Example 2: Use the second derivative to find the relative extrema and points of inflection, if appropriate, for
the function: f ( x ) = x 3 ! 12 x ! 5
Example 3: Use the second derivative to find the relative extrema and points of inflection, if appropriate, for
x
the function: f ( x ) = 2
x +1
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