1. Derivative Sign Patterns for Infinitely
Differentiable Functions in Three-Dimensions
Madeline Edwards
March 9, 2015
Abstract
A derivative sign pattern (DSP) is a sequence of positive and negative
signs that represent the signs of a function and its derivatives over its do-
main. Some infinitely differentiable functions have sign patterns, but not
all. Functions that fall into this category are trigonometric functions, ex-
ponential functions, logarithmic functions, and possibly others. Of these,
only exponential functions have sign patterns; the others take different
signs on their domains. As seen in Calculus I, certain functions can be
differentiated without eventually becoming 0. In the one-dimensional case
studied by Clark, ex
is an example of a function that has an infinite num-
ber of derivatives. In the domain of all real numbers, R, Clark determined
the DSPs and found example functions to match the pattern. In R, Clark
found only four valid sign patterns, all positives, positive and negative
signs alternating, and their negations. In the case of R, a function that
is infinitely differentiable that has a sign pattern can be determined from
the original function and the first derivative. Schilling expanded from the
one-dimensional case to the two-dimensional case for the entire plane. In
the domain of R × R with ordered pairs, Schilling found eight possible
DSPs. Building on Schilling’s Derivative Sign Pattern Theorem, the ex-
pansion to the three-dimensional case is analyzed. The specific case of
interest in three-dimensions is ordered triples of real numbers, R × R × R.
From Schilling’s research of matrix possibilities in two-dimensions, anal-
ysis of what is possible in three-dimensions can be constructed. In the
three-dimensional case, there is interesting geometry among the deriva-
tive sign patterns with only a finite number of possible combinations found
in R×R×R. While applications of DSPs in three-dimensions are limited,
the gained understanding of how functions and derivatives work within a
given domain is the greater purpose to this research.
Derivative Sign Patterns in domain R
In the one-dimensional case from Clark’s research and two-dimensional case from
Schilling’s research, there are only a finite number of derivative sign patterns
(DSP) in the domain of all real numbers, R. When determining all possible DSP
1

2. in three-dimensions, Clark found two possible DSPs and their negations [?] and
Schilling stated that there are four possible DSPs in two-dimensions, and their
negations [?]. By looking at combinations of DSP types, DSP A, DSP B, DSP
C, DSP D, and their negations, with single combinations calculations, there are
56 possible distinct arrangements of two-dimensional DSP in three-dimensions.





+ + + ...
+ + + ...
+ + + ...
...
...
...





(a)





+ − + ...
+ − + ...
+ − + ...
...
...
...





(b)





+ + + ...
− − − ...
+ + + ...
...
...
...





(c)





+ − + ...
− + − ...
+ − + ...
...
...
...





(d)
Figure 1: The valid two-dimensional DSP found by Schilling in R; types DSP A, DSP B, DSP C,
and DSP D.
Valid arrangements of DSPs are where positive and negative signs align with
every slice in three-dimensions with respect to the xy, xz, and yz planes. From
observing all possible combinations in three-dimensions of two-dimensional ar-
rangements, only 8 possible arrangements are valid in addition to their nega-
tions.
DSP Example Function Negation DSP Example Function
A, A, A ex+y+z
Neg. A, Neg. A, Neg. A −ex+y+z
C, C, A e−x+y+z
Neg. C, Neg. C, Neg. A −e−x+y+z
A, B, B ex+y−z
Neg. A, Neg. B, Neg. B −ex+y−z
C, D, B e−x+y−z
Neg. C, Neg. D, Neg. B −e−x+y−z
B, A, C ex−y+z
Neg. B, Neg. A, Neg. C −ex−y+z
D, C, C e−x−y+z
Neg. D, Neg. C, Neg. C −e−x−y+z
B, B, D ex−y−z
Neg. B, Neg. B, Neg. D −ex−y−z
D, D, D e−x−y−z
Neg. D, Neg. D, Neg. D −e−x−y−z
Table 1
The combinations from Schilling’s two-dimensional
cases of derivative sign patterns.
A single equation in one position is either positive or negative. At any given
position in three-dimension, if one perspective of a plane in a single position is
one sign, while another perspective of a different plane, but in the same position,
are of opposite signs, this shows a contradiction. Since a function cannot be both
positive and negative within one position, this contradiction shows an invalid
combination of a three-dimensional DSP.
To represent valid and invalid DSP’s, Figure 2 and Figure 3 display positive
and negative sign values with respect to x, y, z.
2

3. Figure 2: Valid DSP: B, A, C Figure 3: Invalid DSP: Negation B, A, C
The small grid works by creating cubes within the figure represent a single
position with a derivative taken with respect to x, y, and z. The different posi-
tions within the cube along the x, y, z axes demonstrate the various derivatives
taken with respect to each variable. If in a given sub-cube in the figure does
not have identical signs on all faces of the sub-cube, then the DSP is not valid
because there are conflicting signs within a single position.
Derivative Sign Pattern Determination
The derivative sign pattern is determined by knowing the sign pattern of the
original function and its first derivative. As seen in Clark’s research, if (an)
has a sign pattern in the domain R, then an = an+2 for all n. Thus the sign
pattern is completely determined after the original function and first derivative.
In two-dimensions, Schilling’s matrices the exhibit the DSPs are determined by
knowing the original function f(x, y), and the first derivatives, fx, fy. The sign
of fx,y is forced based on the other signs of the first and original functions signs.



+ + ...
+ fx,y ...
...
...


 →



+ + ...
+ − ...
...
...



Figure 4
The invalid two-dimensional DSP found by Schilling in R; type E where this is not
possible, thus the fx,y is forced based on the other sign entries.
Specifically in the matrices, only the first 2 × 2 inputs are needed to know
how the DSPs will behave in R.
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4. Three-Dimensional Derivative Sign Pattern Determination
In three-dimensions, the original function and the first derivative with respect
to x, y, z determine the sign pattern in R. If f(x, y, z) has a sign pattern in R,
then the sign pattern is completely determined by f, fx, fy, and fz.
f fx
fy fx,y
fz fx,z
fy,z fx,y,z
Figure 5
Placing ± signs at any node automatically force surrounding nodes to be a specific sign so
that it does not violate the two-dimensional DSP possibilities listed prior.
This only holds true because in the case of R, there are only a finite number
of patterns that are possible and the patterns remains constant throughout the
entire domain, thus fn = fn+2 ∀n ∈ N, to which n is the power that f is derived.
Thus, like in the one-dimensional case and two-dimensional case, the original
function and first derivative determine the DSP. All even and odd derivatives
have the same sign, respectively. Similarly to the two-dimensional case, in three
dimensions, the signs of fx,y, fx,z, fy,z, and fx,y,z are forced.
4

5. ± ±
± fx,y
± fx,z
fy,z fx,y,z
Figure 6
The known signs of f, fx, fy, fz force the signs of fx,y, fy,z, fx,z, and fx,y,z.
Assume that f, fx, fy, fz signs are known. By strictly looking at the x, y
plane, this forces fx,y to chose a specific sign based on the signs of f, fx, fy, fz
as this two-dimensional perspective must follow one of the valid DSP cases, A,
B, C, D or their negations. Now that fx,y is determined, fx,z is also forced. By
looking at the x, z plane, since f, fx, and fz are known, then fx,z is forced and
must follow a two-dimensional DSP cases. This then leads fx,y,z to be forced
because fx, fx,y, and fx,z are known. We are only left with the last determined
node, fy,z which is forced since fy, fx,y and fx,y,z are known. To determine each
node’s sign with respect to the specific derivative, the surround three nodes of a
single surface must be known. Thus determination follows this pattern strictly,
Known Signs Determines Planes of Cube
f, fx, fy, fz fx,y, fx,z x, y and x, z
f, fx, fy, fz, fx,y, fx,z fx,y,z x, y, z
f, fx, fy, fz, fx,y, fx,z, fx,y,z fy,z y, z
Table 2
Once three node signs are know, a single plane is determined.
Like the one and two-dimensional case, the DSPs in three dimensions is com-
pletely determined by the original function and the first derivative. The first
2 × 2 × 2 cube determines the entire DSP pattern in R × R × R. From the table
above, there are only 16 possibilities of valid DSPs in three-dimensions. Notice
that every example function matching a unique DSP is of the form ±e±x±y±z
.
With each node on the cube, there are two possibilities; either positive or nega-
tive signs. Since there are eight nodes on the 2 × 2 × 2 cube, there are a total of
16 choices of signs on the cube, representing the only possible 16 DSP found in
three dimensions. Looking at the example equations, ±e±x±y±z
, there are 24
5

6. possible combinations, or 16 total possibilities for the function, each uniquely
identifying with only one specific DSP.
Extensions of Research
Possible extensions of this research would be to look at other infinitely differ-
entiable functions that exhibit theses DSPs in derivative sign patterns. While
exponential functions have been noted previously in the table, it is possible that
there are many other types of functions with a DSP in three-dimensions. An-
other area of research would be to look at other domains and possible DSPs.
Domains include the unit cube, R+
× R+
× R+
, (0, 1) × R+
× R+
, as well as
others.
References
[1] Clark, Jeffrey. Derivative Sign Patterns, The College Mathemat-
ics Journal 42.5 (2011): 379-82. JSTOR. Web.20 Sept. 2014.
http://www.jstor.org/stable/10.4169/college.math.j.42.5.379?ref=no-x-
route453a5d25a141997a8acf14a65956bcf4
[2] Schilling, Kenneth. Derivative Sign Patterns in Two Dimensions. The Col-
lege Mathematics Journal 44.2 (2013): 102-08. JSTOR. Web. 20 Sept.
2014. http://www.jstor.org/stable/10.4169/college.math.j.44.2.102?ref=no-
x-route:937a49d3d27b9e5b683e99af4239ce50.
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