The First and Second Derivative Tests
The effect of f ′ on the graph of f
Test for increasing / decreasing:
a. If f ′(x) > 0 on an interval, then f is increasing on the interval.
b. If f ′(x) < 0 on an interval, then f is decreasing on the interval.
The First Derivative Test (for local maximum / minimum)
Suppose c is a critical point of a continuous function f.
a. If f ′ changes from positive (on the left of c) to negative (on the right of c)
at c, then f has a local maximum at c.
b. If f ′ changes from negative to positive at c, then f has a local minimum
at c.
c. If f ′ does not change sign at c (i.e., f ′ remain either positive or negative
on both sides of c), then f has neither a local maximum nor a minimum at c.
Ex. f (x) = x2 + 5 has a local minimum at x = 0.
Ex. f (x) = x3 has no local max or min point, even though it has a critical
point at x = 0.
Ex. Find the local extreme points of h(x) = 2x3 − 18x.
Ex. Find the local extreme points of f (x) = (x + 1)3(x − 1)2.
The effect of f ″ on the graph of f
Definition: Concavity
If the graph of f lies above all of its tangent lines on an interval I, then
it is called concave upward on I. If the graph of f lies below all of its
tangent lines on I, then it is called concave downward on I.
Equivalently,
f is concave upward on I if f ″(x) > 0 for all x on I.
f is concave downward on I if f ″(x) < 0 for all x on I.
Definition: Inflection Point (Point of Inflection or Flex Point)
A point P on a curve y = f (x) is called an inflection point if f is
continuous there and if the concavity of the curve changes there (from
upward to downward, or from downward to upward).
Note: The candidates for inflection points are points where either
f ″ = 0 or f ″ is undefined. However, just like that not every critical
point is a local max / min, not every such point is an inflection point
either. They are just the pool of points you need to check in order to
find the inflection point(s) of a curve.
Comment: It often happens that a graph has different concavity on the
two sides of a vertical asymptote. However, because a curve is not
continuous at a vertical asymptote, it can never have an inflection
point there, even if f is defined there.
Ex. f (x) = x3 has an inflection point at x = 0. In fact, f (x) = x5, f (x) = x7,
f (x) = x9, … , all have an inflection point at x = 0.
Ex. f (x) = x1/3 also has an inflection point at x = 0, even though f ″ is
undefined there. (Therefore, inflection points can exist where f ″ doesn’t.)
Ex. f (x) = x4 does not have an inflection point at x = 0, even though f ″ = 0
there. (Therefore, the fact that f ″ = 0 does not guarantee that an inflection
point exists there.)
The Second Derivative Test (for local maximum / minimum)
Suppose f ″ is continuous near c.
a. If f ′(c) = 0 and f ″(c) > 0, then f has a local minimum at c.
b. If f ′(c) = 0 and f ″(c) < 0, then f has a local maximum at c.
Else, the test fails (if f ′(c) doesn’t exist, or f ″(c) = 0, or f ″(c) doesn’t exist).
Note: Even though it is often easier to use than the first derivative test,
the second derivative test can fail at some points, as noted above. If
the second derivative test fails, then the first derivative test must be
used to classify the point in question.
Ex. f (x) = x2 has a local minimum at x = 0.
Ex. f (x) = x4 has a local minimum at x = 0. But the second derivative test
would fail for this function, because f ″(0) = 0. The first derivative test gives
the correct result.
Ex. Use the second derivative test to find the local extreme points of
h(x) = 2x3 − 18x.
Comment: It’s important to remember that in the first derivative test we
check the intervals between critical points, by evaluate f ′ at some test point
in each interval. While in the second derivative test we check the critical
points themselves (those where f ′ = 0), by evaluate f ″ at each critical point.