Derivative test

Incalculus,aderivative testuses thederivativesof afunctionto locate thecritical pointsof a function and determine whether each point is alocal maximum,alocal minimum,or asaddle point.Derivative tests can also give information about theconcavityof a function.

The usefulness of derivatives to findextremais proved mathematically byFermat's theorem of stationary points.

First-derivative test

The first-derivative test examines a function'smonotonicproperties (where the function isincreasing or decreasing), focusing on a particular point in itsdomain.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 aresufficient conditionsthat 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 thatfis areal-valued function defined on someopen intervalcontaining the pointxand suppose further thatfiscontinuousatx.

• If there exists a positive numberr> 0 such thatfis weakly increasing on(x−r,x]and weakly decreasing on[x,x+r),thenfhas a local maximum atx.
• If there exists a positive numberr> 0 such thatfis strictly increasing on(x−r,x]and strictly increasing on[x,x+r),thenfis strictly increasing on(x−r,x+r)and does not have a local maximum or minimum atx.

Note that in the first case,fis not required to be strictly increasing or strictly decreasing to the left or right ofx,while in the last case,fis 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 aconstant functionis 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 themean value theorem.It is a direct consequence of the way thederivativeis defined and its connection to decrease and increase of a function locally, combined with the previous section.

Supposefis a real-valued function of a real variable defined on someintervalcontaining the critical pointa.Further suppose thatfiscontinuousataanddifferentiableon some open interval containinga,except possibly ataitself.

• If there exists a positive numberr> 0 such that for everyxin (a−r,a) we havef′(x) ≥ 0,and for everyxin (a,a+r) we havef′(x) ≤ 0,thenfhas a local maximum ata.
• If there exists a positive numberr> 0 such that for everyxin (a−r,a) we havef′(x) ≤ 0,and for everyxin (a,a+r) we havef′(x) ≥ 0,thenfhas a local minimum ata.
• If there exists a positive numberr> 0 such that for everyxin (a−r,a) ∪ (a,a+r) we havef′(x) > 0,thenfis strictly increasing ataand 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 notvacuous;there are functions that satisfy none of the first three conditions, e.g.f(x) =x² 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 third, strict inequality is required.

Applications

The first-derivative test is helpful in solvingoptimization problemsin physics, economics, and engineering. In conjunction with theextreme value theorem,it can be used to find the absolute maximum and minimum of a real-valued function defined on aclosedandboundedinterval. In conjunction with other information such as concavity, inflection points, andasymptotes,it can be used to sketch thegraphof a function.

Second-derivative test (single variable)

After establishing thecritical pointsof a function, thesecond-derivative testuses the value of thesecond derivativeat those points to determine whether such points are a localmaximumor a localminimum.If the functionfis twice-differentiableat a critical pointx(i.e. a point wheref′(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 Theoremmay sometimes be used to determine the behavior offnearxusinghigher 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

$0<f''(x)=\lim _{h\to 0}{\frac {f'(x+h)-f'(x)}{h}}=\lim _{h\to 0}{\frac {f'(x+h)}{h}}.$

Thus, forhsufficiently small we get

${\frac {f'(x+h)}{h}}>0,$

which means that$f'(x+h)<0$if$h<0$(intuitively,fis decreasing as it approaches$x$from the left), and that$f'(x+h)>0$if$h>0$(intuitively,fis increasing as we go right fromx). Now, by thefirst-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 isconcave uporconcave downat a point. It does not, however, provide information aboutinflection points.Specifically, a twice-differentiable functionfis 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

Thehigher-order derivative testorgeneral derivative testis 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 ofn= 1 in the higher-order derivative test.

Letfbe a real-valued, sufficientlydifferentiable functionon an interval$I\subset \mathbb {R}$,let$c\in I$,and let$n\geq 1$be anatural number.Also let all the derivatives offatcbe zero up to and including then-th derivative, but with the (n+ 1)th derivative being non-zero:

$f'(c)=\cdots =f^{(n)}(c)=0\quad {\text{and}}\quad f^{(n+1)}(c)\neq 0.$

There are four possibilities, the first two cases wherecis an extremum, the second two wherecis a (local) saddle point:

• Ifnisoddand$f^{(n+1)}(c)<0$,thencis a local maximum.
• Ifnis odd and$f^{(n+1)}(c)>0$,thencis a local minimum.
• Ifnisevenand$f^{(n+1)}(c)<0$,thencis a strictly decreasing point of inflection.
• Ifnis even and$f^{(n+1)}(c)>0$,thencis a strictly increasing point of inflection.

Sincenmust be either odd or even, this analytical test classifies any stationary point off,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.

$f'(x)=6x^{5}$,$f'(0)=0;$

$f''(x)=30x^{4}$,$f''(0)=0;$

$f^{(3)}(x)=120x^{3}$,$f^{(3)}(0)=0;$

$f^{(4)}(x)=360x^{2}$,$f^{(4)}(0)=0;$

$f^{(5)}(x)=720x$,$f^{(5)}(0)=0;$

$f^{(6)}(x)=720$,$f^{(6)}(0)=720.$

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. Thusn= 5, and by the test, there is a local minimum at 0.

Multivariable case

Main article:Second partial derivative test

For a function of more than one variable, the second-derivative test generalizes to a test based on theeigenvaluesof the function'sHessian matrixat the critical point. In particular, assuming that all second-order partial derivatives offare continuous on aneighbourhoodof a critical pointx,then if the eigenvalues of the Hessian atxare all positive, thenxis a local minimum. If the eigenvalues are all negative, thenxis a local maximum, and if some are positive and some negative, then the point is asaddle point.If the Hessian matrix issingular,then the second-derivative test is inconclusive.

See also

• Fermat's theorem (stationary points)
• Maxima and minima
• Karush–Kuhn–Tucker conditions
• Phase line– virtually identical diagram, used in the study of ordinary differential equations
• Bordered Hessian
• Optimization (mathematics)
• Differentiability
• Convex function
• Second partial derivative test
• Saddle point
• Inflection point
• Stationary point

Further reading

• Chiang, Alpha C.(1984).Fundamental Methods of Mathematical Economics(Third ed.). New York: McGraw-Hill. pp.231–267.ISBN0-07-010813-7.
• Marsden, Jerrold;Weinstein, Alan(1985).Calculus I(2nd ed.). New York: Springer. pp. 139–199.ISBN0-387-90974-5.
• Shockley, James E. (1976).The Brief Calculus: with Applications in the Social Sciences(2nd ed.). New York: Holt, Rinehart & Winston. pp. 77–109.ISBN0-03-089397-6.
• Stewart, James(2008).Calculus: Early Transcendentals(6th ed.). Brooks Cole Cengage Learning.ISBN978-0-495-01166-8.
• Willard, Stephen (1976).Calculus and its Applications.Boston: Prindle, Weber & Schmidt. pp. 103–145.ISBN0-87150-203-8.

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 Minimaat Convergence