Sec 4.3 – Monotonic Functions and the First Derivative Test
Monotonicity – defines where a function is increasing or
decreasing.
A function is monotonic if it is increasing or decreasing on an interval.
𝑓 𝑥
a c
d
b
Monotonicity of 𝒇(𝒙)
Interval
Increasing/Decreasing
(𝑎, 𝑐)
𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔
(𝑐, 𝑑)
𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔
(𝑑, 𝑏)
𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔
Sec 4.3 – Monotonic Functions and the First Derivative Test
The First Derivative Test
A function 𝑓 𝑥 is continuous on an open interval containing critical point(s). If
𝑓 𝑥 is differentiable on the interval, except possibly at the critical points, then
𝑓 𝑥 at the critical point(s) can be classified as follows:
1. Local Maximum if 𝑓 𝑥 changes from positive to negative at m.
2. Local Minimum if 𝑓 𝑥 changes from negative to positive at n.
3. If there is no sign change, then the critical point is not a local minimum or
maximum.


𝑓 𝑥

a e m
f
n
g
b
Sec 4.3 – Monotonic Functions and the First Derivative Test
The First Derivative Test


𝑓 𝑥

a e m
f
n
f(critical point)
Extrema
𝑓 𝑚
𝑙𝑜𝑐𝑎𝑙 𝑚𝑎𝑥
𝑓 𝑛
𝑙𝑜𝑐𝑎𝑙 𝑚𝑖𝑛
Test Point
f’(test point)
f’(x)
Inc/Dec
e
𝑓 𝑒 = +
𝑓 𝑥 > 0
Inc.
f
g
𝑓 𝑓 = −
𝑓 𝑔 = +
𝑓 𝑥 < 0
𝑓 𝑥 > 0
g
b
Dec.
Inc.
Sec 4.3 – Monotonic Functions and the First Derivative Test
Example Problems
𝑓 𝑥 = 𝑥 3 − 6𝑥 − 3
𝑓 𝑥 = 𝑥 4 − 6𝑥 2 + 2
𝑓 𝑥 = 𝑥(𝑙𝑛𝑥)2
𝑓 𝑥 = 𝑥 3 − 4𝑥 2 + 3𝑠𝑖𝑛𝑥
Sec 4.4 – Concavity and Curve Sketching
Concavity – defines the curvature of a function.
A function is concave up on an open interval if 𝑓 𝑥 is increasing on the
interval.
A function is concave down on an open interval if 𝑓 𝑥 is decreasing on the
interval.
Point of Inflection (poi) – the point on the graph where the concavity changes.
𝑓 𝑥

a
c
poi
Concavity of f(x)
b
Interval
Concave up/Concave down
(𝑎, 𝑐)
𝑐𝑜𝑛𝑐𝑎𝑣𝑒 𝑑𝑜𝑤𝑛
(𝑐, 𝑏)
𝑐𝑜𝑛𝑐𝑎𝑣𝑒 𝑢𝑝
Sec 4.4 – Concavity and Curve Sketching
The Second Derivative Test for Concavity
The graph of a twice-differentiable function y = f (x) is:
1. Concave up on any interval where 𝑓 𝑥 > 0, and
2. Concave down on any interval where 𝑓 𝑥 < 0.


𝑓 𝑥

a e
f
g
b
x
f’’(x)
f’’(x)
Concave up/Concave down
e
𝑓 𝑒 = −
𝑓 𝑥 < 0
Concave down
f
𝑓 𝑓 = −
𝑓 𝑥 < 0
Concave down
g
𝑓 𝑔 = +
𝑓 𝑥 > 0
Concave up
Sec 4.4 – Concavity and Curve Sketching
The Second Derivative Test for Local Extrema
If 𝑓 𝑐 = 0 (which makes x = c a critical point) and 𝑓 𝑐 < 0, then f has a local
maximum at x = c.
If 𝑓 𝑐 = 0 (which makes x = c a critical point) and 𝑓 𝑐 > 0, then f has a local
minimum at x = c.
NOTE: If the second derivative is equal to zero (or undefined) then the Second
Derivative Test is inconclusive.

𝑓 𝑥

a
Critical Point
m
n
b
f’’(x)
Concavity
Extrema
m
𝑓 𝑚 = −
Concave down
Local max
n
𝑓 𝑛 = +
Concave up
Local min
Sec 4.4 – Concavity and Curve Sketching
Example Problems
𝑓 𝑥 = 𝑥(𝑙𝑛𝑥)2
𝑓 𝑥 = 𝑥 3 − 4𝑥 2 + 3𝑠𝑖𝑛𝑥
𝑓 𝑥 = 𝑥 4 − 6𝑥 2 + 2
𝑓 𝑥 = 𝑥 3 − 6𝑥 − 3
Sec 4.4 – Concavity and Curve Sketching
Curve Sketching