Editing Derivative test (section)

==Second-derivative test (single variable)==
After establishing the [[critical point (mathematics)|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 [[Maxima and minima|maximum]] or a local minimum.<ref>{{cite book |title=Fundamental Methods of Mathematical Economics|url=https://archive.org/details/fundamentalmetho00chia_782|last=Chiang|first=Alpha C.|author-link=Alpha Chiang|page=[https://archive.org/details/fundamentalmetho00chia_782/page/n233 231]–267|year=1984|editor=McGraw-Hill|isbn=0-07-010813-7}}</ref> If the function ''f'' is twice-differentiable at a critical point ''x'' (i.e. a point where ''{{prime|f}}''(''x'') = 0), then:
* If <math>f''(x) < 0</math>, then <math>f</math> has a local maximum at <math>x</math>.
* If <math>f''(x) > 0</math>, then <math>f</math> has a local minimum at <math>x</math>.
* If <math>f''(x) = 0</math>, the test is inconclusive.

In the last case, [[Taylor's theorem#Taylor's theorem in one real variable|Taylor's theorem]] may sometimes be used to determine the behavior of ''f'' near ''x'' using [[higher derivative]]s.

===Proof of the second-derivative test===
Suppose we have <math>f''(x) > 0</math> (the proof for <math>f''(x) < 0</math> is analogous). By assumption, <math>f'(x) = 0</math>. Then 

: <math>0 < f''(x) = \lim_{h \to 0} \frac{f'(x + h) - f'(x)}{h} = \lim_{h \to 0} \frac{f'(x + h)}{h}.</math>

Thus, for ''h'' sufficiently small we get

: <math>\frac{f'(x + h)}{h} > 0,</math>

which means that <math>f'(x + h) < 0</math> if <math>h < 0</math> (intuitively, ''f'' is decreasing as it approaches <math>x</math> from the left), and that <math>f'(x + h) > 0</math> if <math>h > 0</math> (intuitively, ''f'' is increasing as we go right from ''x''). Now, by the [[first-derivative test]], <math>f</math> has a local minimum at <math>x</math>.

===Concavity test===
A related but distinct use of second derivatives is to determine whether a function is [[Concave function|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 <math>f''(x) > 0</math> and concave down if <math>f''(x) < 0</math>. Note that if <math>f(x) = x^4</math>, then <math>x = 0</math> 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''&nbsp;=&thinsp;1 in the higher-order derivative test.

Let ''f'' be a real-valued, sufficiently differentiable function on an interval <math>I \subset \R</math>, let <math>c \in I</math>, and let <math>n \ge 1</math> be a [[natural number]]. Also let all the derivatives of ''f'' at ''c'' be zero up to and including the ''n''-th derivative, but with the (''n''&nbsp;+&thinsp;1)th derivative being non-zero:

: <math>f'(c) = \cdots =f^{(n)}(c) = 0\quad \text{and}\quad f^{(n+1)}(c) \ne 0.</math>

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+1)'' is [[parity (mathematics)|even]] and <math>f^{(n+1)}(c) < 0</math>, then ''c'' is a local maximum.
* If ''(n+1)'' is even and <math>f^{(n+1)}(c) > 0</math>, then ''c'' is a local minimum.
* If ''(n+1)'' is [[parity (mathematics)|odd]] and <math>f^{(n+1)}(c) < 0</math>, then ''c'' is a strictly decreasing point of inflection.
* If ''(n+1)'' is odd and <math>f^{(n+1)}(c) > 0</math>, then ''c'' is a strictly increasing point of inflection.

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

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

: <math>f'(x) = 6x^5</math>, <math>f'(0) = 0;</math>

: <math>f''(x) = 30x^4</math>, <math>f''(0) = 0;</math>

: <math>f^{(3)}(x) = 120x^3</math>, <math>f^{(3)}(0) = 0;</math>

: <math>f^{(4)}(x) = 360x^2</math>, <math>f^{(4)}(0) = 0;</math>

: <math>f^{(5)}(x) = 720x</math>, <math>f^{(5)}(0) = 0;</math>

: <math>f^{(6)}(x) = 720</math>, <math>f^{(6)}(0) = 720.</math>

As shown above, at the point <math>x = 0</math>, the function <math>x^6 + 5</math> has all of its derivatives at 0 equal to 0, except for the 6th derivative, which is positive. Thus ''n''&nbsp;=&nbsp;5, and by the test, there is a local minimum at 0.