Lecture 6
Monotonicity and Concavity
Math 13200
First Derivatives and Monotonicity
One way of thinking of the derivatives are as rates of change.
• If a function represents position, its derivative is velocity.
• If a function represents profit, its derivative is marginal profit.
• If a function represents population, its derivative is population growth.
• If a function represents electric charge, then its derivative is electric current.
In each case, it is quite important what it means when the derivative is positive or negative.
Let f (t) be a differentiable function, and consider t as time. (For the profit example, t represents
units produced.)
Function f (t) represents
Derivative f 0 (t) represents
f 0 (t) > 0
f 0 (t) < 0
position
velocity
profit
marginal profit
should produce more
should produce less
population
population growth
pop. is growing
pop. is shrinking
electric charge
electric current
direction of movement
direction of current
In these examples, we see how the sign of the derivative indicates whether the original function
increases or decreases.
Definitions
Let f be a function defined on an interval I.
• f (x) is increasing on I if for each pair x, y ∈ I we have
x < y ⇒ f (x) ≤ f (y)
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• f (x) is strictly increasing on I if for each pair x, y ∈ I we have
x < y ⇒ f (x) < f (y)
• f (x) is decreasing on I if for each pair x, y ∈ I, we have
x < y ⇒ f (x) ≥ f (y)
• f (x) is strictly decreasing on I if for each pair x, y ∈ I, we have
x < y ⇒ f (x) > f (y)
• f (x) is (strictly) monotonic if it is either (strictly) increasing or (strictly) decreasing.
We say a
Monotonicty Theorem
Let f be continuous on an interval I and differentiable at every interior point of I. Then if for all
interior points x of I

 0
increasing on I.
f (x) ≥ 0






strictly increasing on I.
f 0 (x) > 0
we conclude that f is


decreasing on I.
f 0 (x) ≤ 0





 0
strictly decreasing on I.
f (x) < 0
Example
Determine where
g(x) =
x
1 + x2
is increasing and decreasing.
Answer: We must first use the quotient rule to find the derivative
(1 + x2 ) − x(2x)
(1 + x2 )2
1 − x2
=
(1 + x2 )2
g 0 (x) =
To find where g is increasing/decreasing, we need to know when g 0 is positive/negative. Since
(1 + x2 )2 is always positive, we just need to know when 1 − x2 is positive and negative. Since
1 − x2 = (1 − x)(1 + x)
is greater than 0 if and only if x < 1 and x > −1, we see that
g(x) increasing
⇔ g 0 (x) > 0
g(x) decreasing
⇔
g 0 (x) < 0
2
⇔
x ∈ (−1, 1)
⇔ x ∈ (−∞, −1) ∪ (1, ∞)
Second Derivatives and Concavity
Another way to view derivatives is as the slope of the tangent line. So f 0 (x) may be viewed as the
slope of the tangent line to f at x. By viewing f 00 as the rate of change of f 0 , we thus see that the
second derivative measures the way the slope of the tangent line changes.
Hence, if f 00 > 0, then f is curving upwards, and f 00 < 0, it is curving down.
Definitions
Let f be a continuous function on I.
• A region R in the x-y plane is called convex if for every 2 points P, Q ∈ R, R contains the
line segment joining P and Q.
• The region above f on I is the set of points P = (x0 , y0 ) in the x-y plane such that x0 ∈ I
and y0 > f (x0 ).
• The region below f on I is the set of points P = (x0 , y0 ) in the x-y plane such that x0 ∈ I
and y0 < f (x0 ).
• f is concave up on I if the region above f on I is convex.
• f is concave down on I if the region below f on I is convex.
• an inflection point is an x-value c such that there is an interval (a, b) containing c with f
concave up (or down) on (a, c) and concave down (or up) on (c, b).
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Illustration
Concavity Theorem
For f twice differentiable on an open interval I, we have
(
(
f 00 (x) > 0
concave up on I
If
for all x ∈ I, then f is
00
f (x) < 0
concave down on I
Example (not done in class)
Where is f (x) = 13 x3 − x2 − x + 4 increasing, decreasing, concave up, and concave down?
Answer: We must find the first and second derivatives.
f 0 (x) = 4x2 − 2x − 1 = 4(x + 12 )(x − 1)
f 00 (x) = 8x − 2 = 8(x − 41 )
Thus, we conclude that
f (x) increasing
⇔
f 0 (x) > 0
⇔
x ∈ (−∞, − 12 ) ∪ (1, ∞)
f (x) decreasing
⇔
f 0 (x) < 0
⇔
x ∈ (− 21 , 1)
f (x) concave up
⇔ f 00 (x) > 0
⇔
x ∈ ( 14 , ∞)
f (x) concave down
⇔ f 00 (x) < 0
⇔
x ∈ (−∞, 14 )
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