Incalculus, aderivative test uses thederivatives of afunction to locate thecritical points of a function and determine whether each point is alocal maximum, alocal minimum, or asaddle point. Derivative tests can also give information about theconcavity of a function.

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

The first-derivative test examines a function'smonotonic properties (where the function is increasing 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 conditions that guarantee the monotonicity properties above, and these conditions apply to the vast majority of functions one would encounter.

Stated precisely, suppose thatf is areal-valued function defined on someopen interval containing the pointx and suppose further thatf iscontinuous atx.

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

Note that in the first case,f is not required to be strictly increasing or strictly decreasing to the left or right ofx, 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 aconstant function is considered both a local maximum and a local minimum.

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 thederivative is defined and its connection to decrease and increase of a function locally, combined with the previous section.

Supposef is a real-valued function of a real variable defined on someinterval containing the critical pointa. Further suppose thatf is continuous ata anddifferentiable on some open interval containinga, except possibly ata itself.

• If there exists a positive numberr > 0 such that for everyx in (a −r,a) we havef′(x) ≥ 0, and for everyx in (a,a +r) we havef′(x) ≤ 0, thenf has a local maximum ata.
• If there exists a positive numberr > 0 such that for everyx in (a −r,a) we havef′(x) ≤ 0, and for everyx in (a,a +r) we havef′(x) ≥ 0, thenf has a local minimum ata.
• If there exists a positive numberr > 0 such that for everyx in (a −r,a) ∪ (a,a +r) we havef′(x) > 0, thenf is strictly increasing ata 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 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.

The first-derivative test is helpful in solvingoptimization problems in 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 aclosed andbounded interval. In conjunction with other information such as concavity, inflection points, andasymptotes, it can be used to sketch thegraph of a function.

After establishing thecritical points of a function, thesecond-derivative test uses the value of thesecond derivative at those points to determine whether such points are a localmaximum or a local minimum.^[1] If the functionf is twice-differentiable at a critical pointx (i.e. a point wheref′(x) = 0), then:

• If, then${\displaystyle f}$ has a local maximum at${\displaystyle x}$.
• If${\displaystyle f^{″}(x)>0}$, then${\displaystyle f}$ has a local minimum at${\displaystyle x}$.
• If${\displaystyle f^{″}(x)=0}$, the test is inconclusive.

In the last case,Taylor's theorem may sometimes be used to determine the behavior off nearx usinghigher derivatives.

Suppose we have${\displaystyle f^{″}(x)>0}$ (the proof for is analogous). By assumption,${\displaystyle f^{\prime }(x)=0}$. Then

Thus, forh sufficiently small we get

${\displaystyle \frac{f^{\prime }(x+h)}{h}>0,}$

which means that if (intuitively,f is decreasing as it approaches${\displaystyle x}$ from the left), and that${\displaystyle f^{\prime }(x+h)>0}$ if${\displaystyle h>0}$ (intuitively,f is increasing as we go right fromx). Now, by thefirst-derivative test,${\displaystyle f}$ has a local minimum at${\displaystyle x}$.

A related but distinct use of second derivatives is to determine whether a function isconcave up or concave down at a point. It does not, however, provide information aboutinflection points. Specifically, a twice-differentiable functionf is concave up if${\displaystyle f^{″}(x)>0}$ and concave down if. Note that if${\displaystyle f(x)=x^4}$, then${\displaystyle x=0}$ 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.

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

Letf be a real-valued, sufficiently differentiable function on an interval${\displaystyle I\subset \mathbb{R}}$, let${\displaystyle c\in I}$, and let${\displaystyle n\ge 1}$ be anatural number. Also let all the derivatives off atc be zero up to and including then-th derivative, but with the (n + 1)th derivative being non-zero:

${\displaystyle f^{\prime }(c)=\cdots =f^{(n)}(c)=0\text{and}{f}^{(n+1)}(c)\ne 0.}$

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

• If(n+1) iseven and, thenc is a local maximum.
• If(n+1) is even and${\displaystyle f^{(n+1)}(c)>0}$, thenc is a local minimum.
• If(n+1) isodd and, thenc is a strictly decreasing point of inflection.
• If(n+1) is odd and${\displaystyle f^{(n+1)}(c)>0}$, thenc is a strictly increasing point of inflection.

Since(n+1) must be either odd or even, this analytical test classifies any stationary point off, so long as a nonzero derivative shows up eventually, where${\displaystyle f^{(n+1)}(c)\ne 0.}$ is the first non-zero derivative.

Say we want to perform the general derivative test on the function${\displaystyle f(x)=x^6+5}$ at the point${\displaystyle x=0}$. To do this, we calculate the derivatives of the function and then evaluate them at the point of interest until the result is nonzero.

${\displaystyle f^{\prime }(x)=6x^5}$,${\displaystyle f^{\prime }(0)=0;}$

${\displaystyle f^{″}(x)=30x^4}$,${\displaystyle f^{″}(0)=0;}$

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

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

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

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

As shown above, at the point${\displaystyle x=0}$, the function${\displaystyle x^6+5}$ 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.

For a function of more than one variable, the second-derivative test generalizes to a test based on theeigenvalues of the function'sHessian matrix at the critical point. In particular, assuming that all second-order partial derivatives off are continuous on aneighbourhood of a critical pointx, then if the eigenvalues of the Hessian atx are all positive, thenx is a local minimum. If the eigenvalues are all negative, thenx is 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.

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• "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

• Articles with short description
• Short description matches Wikidata