In
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, a derivative test uses the
derivatives of a
function to locate the
critical points of a function and determine whether each point is a
local maximum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...
, a
local minimum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...
, or a
saddle point. Derivative tests can also give information about the
concavity
In calculus, the second derivative, or the second order derivative, of a function is the derivative of the derivative of . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, ...
of a function.
The usefulness of derivatives to find
extrema is proved mathematically by
Fermat's theorem of stationary points.
First-derivative test
The first-derivative test examines a function's
monotonic properties (where the function is
increasing or decreasing), focusing on a particular point in its
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...
. If the function "switches" from increasing to decreasing at the point, then the function will achieve a highest value at that point. Similarly, if the function "switches" from decreasing to increasing at the point, then it will achieve a least value at that point. If the function fails to "switch" and remains increasing or remains decreasing, then no highest or least value is achieved.
One can examine a function's monotonicity without calculus. However, calculus is usually helpful because there are
sufficient conditions that guarantee the monotonicity properties above, and these conditions apply to the vast majority of functions one would encounter.
Precise statement of monotonicity properties
Stated precisely, suppose that ''f'' is a
real-valued function defined on some
open interval containing the point ''x'' and suppose further that ''f'' is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
at ''x''.
* If there exists a positive number ''r'' > 0 such that ''f'' is weakly increasing on and weakly decreasing on , then ''f'' has a local maximum at ''x''.
* If there exists a positive number ''r'' > 0 such that ''f'' is strictly increasing on and strictly increasing on , then ''f'' is strictly increasing on and does not have a local maximum or minimum at ''x''.
Note that in the first case, ''f'' is not required to be strictly increasing or strictly decreasing to the left or right of ''x'', while in the last case, ''f'' is required to be strictly increasing or strictly decreasing. The reason is that in the definition of local maximum and minimum, the inequality is not required to be strict: e.g. every value of a
constant function is considered both a local maximum and a local minimum.
Precise statement of first-derivative test
The first-derivative test depends on the "increasing–decreasing test", which is itself ultimately a consequence of the
mean value theorem. It is a direct consequence of the way the
derivative is defined and its connection to decrease and increase of a function locally, combined with the previous section.
Suppose ''f'' is a real-valued function of a real variable defined on some
interval containing the critical point ''a''. Further suppose that ''f'' is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
at ''a'' and
differentiable on some open interval containing ''a'', except possibly at ''a'' itself.
* If there exists a positive number ''r'' > 0 such that for every ''x'' in (''a'' − ''r'', ''a'') we have and for every ''x'' in (''a'', ''a'' + ''r'') we have then ''f'' has a local maximum at ''a''.
* If there exists a positive number ''r'' > 0 such that for every ''x'' in (''a'' − ''r'', ''a'') ∪ (''a'', ''a'' + ''r'') we have then ''f'' is strictly increasing at ''a'' and has neither a local maximum nor a local minimum there.
* If none of the above conditions hold, then the test fails. (Such a condition is not
vacuous; there are functions that satisfy none of the first three conditions, e.g. ''f''(''x'') = ''x''
2 sin(1/''x'')).
Again, corresponding to the comments in the section on monotonicity properties, note that in the first two cases, the inequality is not required to be strict, while in the next two, strict inequality is required.
Applications
The first-derivative test is helpful in solving
optimization problems in physics, economics, and engineering. In conjunction with the
extreme value theorem, it can be used to find the absolute maximum and minimum of a real-valued function defined on a
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
and
bounded interval. In conjunction with other information such as concavity, inflection points, and
asymptotes, it can be used to sketch the
graph of a function.
Second-derivative test (single variable)
After establishing the
critical points of a function, the ''second-derivative test'' uses the value of the
second derivative at those points to determine whether such points are a local
maximum or a local
minimum.
If the function ''f'' is twice-
differentiable at a critical point ''x'' (i.e. a point where '(''x'') = 0), then:
* If
, then
has a local maximum at
.
* If
, then
has a local minimum at
.
* If
, the test is inconclusive.
In the last case,
Taylor's Theorem may sometimes be used to determine the behavior of ''f'' near ''x'' using
higher derivatives.
Proof of the second-derivative test
Suppose we have
(the proof for
is analogous). By assumption,
. Then
:
Thus, for ''h'' sufficiently small we get
:
which means that
if
(intuitively, ''f'' is decreasing as it approaches
from the left), and that
if
(intuitively, ''f'' is increasing as we go right from ''x''). Now, by the
first-derivative test,
has a local minimum at
.
Concavity test
A related but distinct use of second derivatives is to determine whether a function is
concave up or
concave down at a point. It does not, however, provide information about
inflection points
In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of ...
. Specifically, a twice-differentiable function ''f'' is concave up if
and concave down if
. Note that if
, then
has zero second derivative, yet is not an inflection point, so the second derivative alone does not give enough information to determine whether a given point is an inflection point.
Higher-order derivative test
The ''higher-order derivative test'' or ''general derivative test'' is able to determine whether a function's critical points are maxima, minima, or points of inflection for a wider variety of functions than the second-order derivative test. As shown below, the second-derivative test is mathematically identical to the special case of ''n'' = 1 in the higher-order derivative test.
Let ''f'' be a real-valued, sufficiently
differentiable function on an interval
, let
, and let
be a
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
. Also let all the derivatives of ''f'' at ''c'' be zero up to and including the ''n''-th derivative, but with the (''n'' + 1)th derivative being non-zero:
:
There are four possibilities, the first two cases where ''c'' is an extremum, the second two where ''c'' is a (local) saddle point:
* If ''n'' is
odd
Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric.
Odd may also refer to:
Acronym
* ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
and
, then ''c'' is a local maximum.
* If ''n'' is odd and
, then ''c'' is a local minimum.
* If ''n'' is
even
Even may refer to:
General
* Even (given name), a Norwegian male personal name
* Even (surname)
* Even (people), an ethnic group from Siberia and Russian Far East
**Even language, a language spoken by the Evens
* Odd and Even, a solitaire game wh ...
and
, then ''c'' is a strictly decreasing point of inflection.
* If ''n'' is even and
, then ''c'' is a strictly increasing point of inflection.
Since ''n'' must be either odd or even, this analytical test classifies any stationary point of ''f'', so long as a nonzero derivative shows up eventually.
Example
Say we want to perform the general derivative test on the function
at the point
. To do this, we calculate the derivatives of the function and then evaluate them at the point of interest until the result is nonzero.
:
,
:
,
:
,
:
,
:
,
:
,
As shown above, at the point
, the function
has all of its derivatives at 0 equal to 0, except for the 6th derivative, which is positive. Thus ''n'' = 5, and by the test, there is a local minimum at 0.
Multivariable case
For a function of more than one variable, the second-derivative test generalizes to a test based on the
eigenvalues of the function's
Hessian matrix at the critical point. In particular, assuming that all second-order partial derivatives of ''f'' are continuous on a
neighbourhood of a critical point ''x'', then if the eigenvalues of the Hessian at ''x'' are all positive, then ''x'' is a local minimum. If the eigenvalues are all negative, then ''x'' is a local maximum, and if some are positive and some negative, then the point is a
saddle point. If the Hessian matrix is
singular
Singular may refer to:
* Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms
* Singular homology
* SINGULAR, an open source Computer Algebra System (CAS)
* Singular or sounder, a group of boar ...
, then the second-derivative test is inconclusive.
See also
*
Fermat's theorem (stationary points)
*
Maxima and minima
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...
*
Karush–Kuhn–Tucker conditions
In mathematical optimization, the Karush–Kuhn–Tucker (KKT) conditions, also known as the Kuhn–Tucker conditions, are first derivative tests (sometimes called first-order necessary conditions) for a solution in nonlinear programming to be o ...
*
Phase line – virtually identical diagram, used in the study of ordinary differential equations
*
Bordered Hessian
In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed ...
*
Optimization (mathematics)
*
Differentiability
*
Convex function
*
Second partial derivative test
The second (symbol: s) is the unit of time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally to 60 seconds each ...
*
Saddle point
*
Inflection point
In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case ...
*
Stationary point
Further reading
*
*
*
*
*
References
{{reflist
External links
"Second Derivative Test" at MathworldConcavity and the Second Derivative TestThomas Simpson's use of Second Derivative Test to Find Maxima and Minimaat Convergence
Differential calculus