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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 f''(x) < 0, then f has a local maximum at x. * If f''(x) > 0, then f has a local minimum at x. * If f''(x) = 0, 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 f''(x) > 0 (the proof for f''(x) < 0 is analogous). By assumption, f'(x) = 0. Then : 0 < f''(x) = \lim_ \frac = \lim_ \frac. Thus, for ''h'' sufficiently small we get : \frac > 0, which means that f'(x + h) < 0 if h < 0 (intuitively, ''f'' is decreasing as it approaches x from the left), and that f'(x + h) > 0 if h > 0 (intuitively, ''f'' is increasing as we go right from ''x''). Now, by the first-derivative test, f has a local minimum at x.


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 f''(x) > 0 and concave down if f''(x) < 0. Note that if f(x) = x^4, then x = 0 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 I \subset \R, let c \in I, and let n \ge 1 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: : f'(c) = \cdots =f^(c) = 0\quad \text\quad f^(c) \ne 0. 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 f^(c) < 0, then ''c'' is a local maximum. * If ''n'' is odd and f^(c) > 0, 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 f^(c) < 0, then ''c'' is a strictly decreasing point of inflection. * If ''n'' is even and f^(c) > 0, 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 f(x) = x^6 + 5 at the point x = 0. To do this, we calculate the derivatives of the function and then evaluate them at the point of interest until the result is nonzero. : f'(x) = 6x^5, f'(0) = 0; : f''(x) = 30x^4, f''(0) = 0; : f^(x) = 120x^3, f^(0) = 0; : f^(x) = 360x^2, f^(0) = 0; : f^(x) = 720x, f^(0) = 0; : f^(x) = 720, f^(0) = 720. As shown above, at the point x = 0, the function x^6 + 5 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 Mathworld

Concavity and the Second Derivative Test

Thomas Simpson's use of Second Derivative Test to Find Maxima and Minima
at Convergence Differential calculus