Section 5.2: Monotonicity and Concavity
What does the derivative, f 0 , say about a function f ?
• If f 0 (x) > 0 on some interval, then f is increasing on that interval.
• If f 0 (x) < 0 on some interval, then f is decreasing on that interval.
An increasing or decreasing function is called monotonic or monotone.
What does the second derivative, f 00 , say about a function f ?
• If f 00 (x) > 0 on some interval, then f is concave upward on that interval.
• If f 00 (x) < 0 on some interval, then f is concave downward on that interval.
A point where a curve changes its direction of concavity is called an inflection point.
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Example: The graph of the derivative f 0 of a function f is given below.
(a) On what intervals is f increasing or decreasing?
(b) On what intervals is f concave upward or downward?
(c) State the x-coordinates of the points of inflection.
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Example: Determine the intervals on which each function is increasing, decreasing, concave
up, or concave down.
(a) f (x) = x2 − x + 3
(b) f (x) = 1 − 3x + 5x2 − x3
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(c) f (x) =
x2
x2 + 1
(d) f (x) = e−x
2 /2
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Example: (Logistic Growth) Suppose the size of a population at time t ≥ 0 is N (t) and
dN
= rN
dt
N
1−
= f (N )
K
where r and K are positive constants.
(a) Compute f 0 (N ) and determine where f is increasing or decreasing.
(b) If r = 3 and K = 10, determine where f is increasing or decreasing.
(c) Graph the growth rate f (N ) as a function of the population size N .
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Example: (Predation) Spruce budworms are a major pest that defoliates balsam fir. They
are preyed upon by birds. A model for the per capita predation rate is
f (N ) =
k2
aN
+ N2
where N denotes the density of budworms and a and k are positive constants.
(a) Find f 0 (N ) and determine where the predation rate is increasing or decreasing.
(b) If a = 5 and k = 10, determine where f is increasing or decreasing.
(c) Graph the growth rate f (N ) as a function of the budworm density N .
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