Dr. Byrne Spring 2012
MA125: Comparing 1st and 2nd Derivative Tests
Definition of a critical point:
A number c in the domain of f (x) is called a critical point if f '(c) = 0 or f '(c) does not exist.
Comments:  The critical points of f (x) are always defined by the zeros (or undefined points) of the
first derivative.
 In order to be a critical point, a point p must be in the domain of f (x). Often points for
which or f '(x) are not defined are also points f (x) where is not defined, so we would
exclude these.
1st Derivative Test
Let f(x) be differentiable. If c is a critical point, then
 if f '(x) changes from + to − at c, f (c) is a local max.
 if f '(x) changes from − to + at c, f (c) is a local min.
2nd Derivative Test
Let f(x) be differentiable. If c is a critical point, and f ''(x) exists then
 f ''(x) > 0 => f (c) is a local minimum
 f ''(x) < 0 => f (c) is a local maximum
 f ''(x) = 0 => inconclusive (f (c) may be local max, local min or neither)
Comment: if the concavity changes and f ''(x)=0 (exists), f (c) is a point of inflection
Example:
Comparing the 1st and 2nd derivative tests for classifying the critical point of 𝑓(𝑥) = 𝑥 2 − 𝑥 − 2.
(a) Find the critical point of f (x).
(b) Use the first derivative test to classify this critical point as a
min, max or neither.
(c) Use the second derivative test to classify this critical point as a min, max or neither.
Example:
Comparing the 1st and 2nd derivative tests for classifying the critical point of 𝑓(𝑥) = 𝑥 3 + 3.
(a) Find the critical point of f (x).
(b) Use the first derivative test to classify this critical point as a
min, max or neither.
(c) Use the second derivative test to classify this critical point as a min, max or neither.
Example:
Comparing the 1st and 2nd derivative tests for classifying the critical point of 𝑓(𝑥) = 14𝑥 4 + 1.
(a) Find the critical point of f (x).
(b) Use the first derivative test to classify this critical point as a
min, max or neither.
(c) Use the second derivative test to classify this critical point as a min, max or neither.