4.2 Concavity
DEFINITION: Concavity of a function
•
Let the function f be differentiable on an interval (a,b). Then
•
1. f is concave upward on (a,b) if f’(x) is increasing on (a,b).
(PALM
UP)
•
f is concave downward on (a,b) if f’(x) is decreasing on (a,b).
(PALM
DOWN)
A point on the graph of a differentiable function at which the concavity
changes
is called an inflection point
Look at the graph of
y = f(x)
Find the largest open interval where f is
concave up.
Find the largest open interval where f is
concave down up.
Find al infection points of f.
THEOREM
•
If f ”(x) > 0 for each value of x in (a,b), then
f is concave upward on (a,b).
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If f ”(x) < 0 for each value of x in (a,b), then
f is concave downward on (a,b).
Procedure for determining the Intervals of Concavity when no graph is
given
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1. Determine the values of x for which
f”(x) = 0 or f”(x) DNE
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2. Identify the open intervals determined by
these numbers.
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3. Determine the sign of f”(x) in each of
these intervals by
computing f”(c) where
c is a test value chosen in each interval.
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•
4. If f”(c) > 0, f is concave upward on (a,b)
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 Spring '08
 VAUGHN
 Calculus, Critical Point, Derivative, Inflection Points, Mathematical analysis, Concave function

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