Dr. Byrne
Math 125 Spring 2015
Name____________________________________
4.3/4.4 Comparing The First and Second Derivative Tests
The first and second derivative tests can be used to identify a critical point as a minimum or a
maximum.
Suppose 𝑓(𝑥) is continuous in a neighborhood containing a critical point c.
1st Derivative Test
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if 𝑓′(𝑥) changes from + to − at c, 𝑓(𝑐) is a local maximum.
if 𝑓 ′ (𝑥) changes from − to + at c, 𝑓(𝑐) is a local minimum.
if 𝑓′(𝑥) does not change sign at c, 𝑓(𝑐) is neither a minimum nor a maximum.
2nd Derivative Test
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if 𝑓 ′′ (𝑥) > 0, 𝑓(𝑐) is a local minimum
if 𝑓 ′′ (𝑥) < 0, 𝑓(𝑐)is a local maximum
if 𝑓 ′′ (𝑥) = 0 or does not exist, the test is inconclusive
(𝑓(𝑐) may be local max, local min or neither)
(Note: An inflection point occurs where 𝑓 ′′ (𝑥) exists and the concavity changes.)
Example A: 𝑦 = 3𝑥 − 𝑥 3
1) On the graph provided, label all maxima,
minima and inflection points (max/min/I).
2) Find all critical points of y:
3) First derivative test:
a. Identify the intervals where y is increasing or decreasing:
y increasing:
y decreasing:
.
.
y concave up:
y concave down:
.
.
b. What does the first derivative test tell us about each critical point?
4) Second derivative test:
a. Identify the intervals where y is concave up or concave down:
b. What does the second derivative test tell us about each critical point?
Example B:
𝑦 = 𝑥 5 − 5𝑥 4
1) On the graph provided, label all maxima,
minima and inflection points (max/min/I).
2) Find all critical points of y:
3) First derivative test:
a. Identify the intervals where y is increasing or decreasing:
y increasing:
y decreasing:
.
.
y concave up:
y concave down:
.
.
b. What does the first derivative test tell us about each critical point?
4) Second derivative test:
a. Identify the intervals where y is concave up or concave down:
b. What does the second derivative test tell us about each critical point?
Example C: 𝑦 = 5𝑥 2/5 − 2𝑥
5) On the graph provided, label all maxima,
minima and inflection points (max/min/I).
6) Find all critical points of y:
7) First derivative test:
a. Identify the intervals where y is increasing or decreasing:
y increasing:
y decreasing:
.
.
y concave up:
y concave down:
.
.
b. What does the first derivative test tell us about each critical point?
8) Second derivative test:
a. Identify the intervals where y is concave up or concave down:
b. What does the second derivative test tell us about each critical point?