x
y
x
y
x
y
f(x) = x2m f(x) = ­x2m f(x) = x2m+1
f(x) = ­x2m+1
MATH 164, Spring 2001 Due Date: Name(s):
Extra Project 12.10b: The Second Derivative Test Again
Objective
One of the major limitations of the Second Derivative Test for functions f : R  R of one real variable is
that if f (0) = 0 (so x = 0 is a critical value for f) and f (0) = 0 then the Second Derivative Test provides
no information about whether f(0) is a relative maximum, a relative minimum, or neither. In this project
we illustrate how we can get more information using concepts related to Mclaurin series.
Narrative
Our analysis is based on Mclaurin series, and the fact that if:
1. f(x) = x2n
, n an integer, then f is positive for all values of x other than x = 0, so f(0) = 0 is a relative
minimum,
2. f(x) = -x2n
, n an integer, then f is negative for all values of x other than x = 0, so f(0) = 0 is a
relative maximum, and
3. f(x) = x2n+1
, n an integer, then the graph of f is positive for some values of x near x = 0 and
negative for others, so f(0) is a neither a relative minimum nor a relative maximum.
The Mclaurin series expansion for a real-valued function f : R  R is:
f(x) =

n=0
f(n)
(0)
n!
xn
= f(0) +
f (0)
1!
x +
f (0)
2!
x2
+
f (0)
3!
x3
+ . . . .
Thus if x = 0 is a critical value for f, then
f(x) = f(0) +
f (0)
2!
x2
+
f (0)
3!
x3
+ . . . .
If we also know that f (0) = 0, then
f(x) = f(0) +
f (0)
3!
x3
+ . . . .
Indeed, suppose that
f(x) = f(0) +
f(N)
(0)
N!
xN
+
f(N+1)
(0)
(N + 1)!
xN+1
+ . . .
where N is the first integer for which f(N)
(0) = 0. Since xN+1
is considerably smaller than xN
for all values
of x sufficiently close to 0, it follows that
f(x)  g(x) = f(0) +
f(N)
(0)
N!
xN
for all values of x sufficiently close to 0. Thus we can classify the behavior of f near 0 by classifying the
behavior of g near 0.
Example: If f(x) = x8
- x3
then f (0) = 0 and f (0) = 0, so the Second Derivative Test provides no
information. However, if we write f(x) = -x3
+ x8
and remember that x8
is considerably smaller than x3
for all values of x sufficiently close to 0, then we may approximate f(x) = -x3
+ x8
by g(x) = -x3
. Since g
has neither a relative maximum nor a relative minimum at x = 0, f has neither a relative maximum nor a
relative minimum at x = 0.
Example: If f(x) = x2
(ex2
-1) then f (0) = 0 and f (0) = 0, so the Second Derivative Test again provides
no information. However, since the Mclaurin series expansion for ex
is
ex
= 1 +
1
1!
x +
1
2!
x2
+ . . . ,
it follows that
ex2
= 1 +
1
1!
x2
+
1
2!
x4
+ . . .
so
f(x) = x2
(ex2
- 1) = x2
1 +
1
1!
x2
+
1
2!
x4
+ . . . - 1 =
1
1!
x4
+
1
2!
x6
+ . . . .
Since x6
and all higher powers of x are considerably smaller than x4
for all values of x sufficiently close to 0,
we may approximate f(x) by g(x) = x4
. Since g has a relative minimum x = 0, f has a relative minimum
at x = 0.
Task
For each of the following functions, f (0) = 0 and f (0) = 0, so the Second Derivative Test provides no
information. Use an analysis of the Mclaurin series of f to determine whether f(0) is a relative maximum,
a relative minimum, or neither.
1. f(x) = 2x6
- 6x4
5. f(x) = 1 - cos x2
2. f(x) = 3x4
- 4x3
+ 6 6. f(x) = x Tan-1
x2
3. f(x) = 3x5
- 5x3
7. f(x) = x - ln(1 + x2
)
4. f(x) = 8x2
- 2x4
8. f(x) = ex3
- 1
Comments
1. The technique presented above can easily be extended to values c of x for which f (c) = 0 and f (c) = 0,
other than c = 0. The only changes that must be made are: 1) shifting attention from the Mclaurin
series expansion for f to the Taylor series expansion for f about x = c, and 2) shifting attention from
values of x that are close to 0 to values of x that are close to c.
2. If, for a given function f, you know that f (0) = 0 and f (0) = 0 but you do not know the Mclaurin
series expansion for f, then you can avoid the intermediate step of finding the Mclaurin series expansion
for f by computing higher and higher order derivatives of f at 0, until you obtain one -- the first one --
at which f(N)
(0) = 0. Depending on whether N is odd or even, and if N is even whether f(N)
(0) > 0 or
f(N)
(0) < 0, you can draw a conclusion about whether f(0) is a relative maximum, a relative minimum,
or neither. (Maple is well suited for this!) This test is a generalization of the Second Derivative Test,
and although we could state it as a higher-order version of the Second Derivative Test, doing so is not
necessarily useful (since the conditions are a little difficult to memorize). Instead, it is perhaps wiser to
simply remember the origins of this test in the context of Mclaurin series.