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
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s of a
function to locate the
critical points of a function and determine whether each point is a
local maximum, a
local minimum, or a
saddle point. Derivative tests can also give information about the
concavity 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
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of ord ...
properties (where the function is
increasing or decreasing), focusing on a particular point in its
domain. 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 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
In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image).
Basic properti ...
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
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
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 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 problem
In mathematics, computer science and economics, an optimization problem is the problem of finding the ''best'' solution from all feasible solutions.
Optimization problems can be divided into two categories, depending on whether the variables ...
s 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 and
bounded
Boundedness or bounded may refer to:
Economics
* Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision
* Bounded e ...
interval. In conjunction with other information such as concavity, inflection points, and
asymptote
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
s, it can be used to sketch the
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
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
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 ...
.
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
In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the ...
may sometimes be used to determine the behavior of ''f'' near ''x'' using
higher derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s.
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. 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 and
, then ''c'' is a local maximum.
* If ''n'' is odd and
, then ''c'' is a local minimum.
* If ''n'' is
even 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
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...
s of the function's
Hessian matrix
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 ...
at the critical point. In particular, assuming that all second-order partial derivatives of ''f'' are continuous on a
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...
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, then the second-derivative test is inconclusive.
See also
*
Fermat's theorem (stationary points)
*
Maxima and minima
*
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 ...
*
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 develo ...
*
Optimization (mathematics)
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
*
Differentiability
*
Convex function
*
Second partial derivative test
*
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
In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" i ...
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