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→What the test says: one-sided sign versions

{{maxmin test}}

==Statement==

The '''first derivative test''' is a partial (i.e., not always conclusive) test used to determine whether a particular [[critical point]] in the [[domain]] of a [[function]] is a point where the function attains a [[local maximum value]], [[local minimum value]], or neither. There are cases where the test is ''inconclusive'', which means that we cannot draw any conclusion.

The one-sided version of this test is also used to determine whether an endpoint of the domain of a function gives an endpoint extremum, and if so, whether it is an endpoint maximum or endpoint minimum.

===What the test says: one-sided sign versions===

{| class="sortable" border="1"

! Continuity and differentiability assumption !!Hypothesis on sign of derivative!! Conclusion!! Prototypical pictures (the dotted point corresponds to <math>(c,f(c))</math>, and the dashed line is the one-sided tangent line at the point)

|-

| <math>f</math> is left continuous ''at'' <math>c</math> and differentiable on the immediate left of <math>c</math> || <math>\! f'(x)</math> is positive (respectively, nonnegative) for <math>x</math> to the immediate left of <math>c</math> (i.e., for <math>x \in (c - \delta, c)</math> for sufficiently small <math>\delta > 0</math>)|| <math>f</math> has a strict local maximum from the left at <math>c</math>, i.e., <math>f(c) > f(x)</math> (respectively, <math>f</math> has a local maximum from the left at <math>c</math>, i.e., <math>f(c) \ge f(x)</math>) for <math>x</math> to the immediate left of <math>c</math>.|| [[File:Leftincreasingconcaveup.png|100px]][[File:Leftincreasingconcavedownflat.png|80px]]

|-

|<math>f</math> is left continuous ''at'' <math>c</math> and differentiable on the immediate left of <math>c</math> ||<math>\! f'(x)</math> is negative (respectively, nonpositive) for <math>x</math> to the immediate left of <math>c</math> (i.e., for <math>x \in (c - \delta, c)</math> for sufficiently small <math>\delta > 0</math>)|| <math>f</math> has a strict local minimum from the left at <math>c</math>, i.e., <math>f(c) < f(x)</math> (respectively, <math>f</math> has a local minimum from the left at <math>c</math>, i.e., <math>f(c) \le f(x)</math>) for <math>x</math> to the immediate left of <math>c</math>.|| [[File:Leftdecreasingconcaveupflat.png|80px]][[File:Leftdecreasingconcavedown.png|100px]]

|-

| <math>f</math> is right continuous ''at'' <math>c</math> and differentiable on the immediate right of <math>c</math> ||<math>\! f'(x)</math> is positive (respectively, nonnegative) for <math>x</math> to the immediate right of <math>c</math> (i.e., for <math>x \in (c,c + \delta)</math> for sufficiently small <math>\delta > 0</math>)|| <math>f</math> has a strict local minimum from the right at <math>c</math>, i.e., <math>f(c) < f(x)</math> (respectively, <math>f</math> has a local minimum from the right at <math>c</math>, i.e., <math>f(c) \le f(x)</math>) for <math>x</math> to the immediate right of <math>c</math>.|| [[File:Rightincreasingconcaveupnotflat.png|100px]][[File:Rightincreasingconcavedown.png|80px]]

|-

| <math>f</math> is right continuous ''at'' <math>c</math> and differentiable on the immediate right of <math>c</math> ||<math>\! f'(x)</math> is negative (respectively, nonpositive) for <math>x</math> to the immediate right of <math>c</math> (i.e., for <math>x \in (c,c + \delta)</math> for sufficiently small <math>\delta > 0</math>)|| <math>f</math> has a strict local maximum from the right at <math>c</math>, i.e., <math>f(c) > f(x)</math> (respectively, <math>f</math> has a local maximum from the right at <math>c</math>, i.e., <math>f(c) \ge f(x)</math>) for <math>x</math> to the immediate right of <math>c</math>.|| [[File:Rightdecreasingconcavedownnotflat.png|100px]][[File:Rightdecreasingconcaveup.png|80px]]

|}

<center>{{#widget:YouTube|id=HZHe9AJAN5s}}</center>

===What the test says: combined sign versions===

{| class="sortable" border="1"

! Continuity and differentiability assumption !! Sign of the derivative <math>f'</math> on immediate left of <math>c</math> !! Sign of <math>f'</math> on immediate right of <math>c</math> !! Conclusion ~~about ~~for <math>f</math> at <math>c</math>: local minimum, local maximum, or neither? !! Prototypical pictures (the dotted point is <math>(c,f(c))</math>)

|-

| <math>f</math> is continuous at <math>c</math> and differentiable on the immediate left and immediate right of <math>c</math> || positive || negative || strict local maximum(two-sided) || [[File:Strictlocalmaxwithderivativezero.png|100px]] [[File:Strictlocalmaxwithundefinedderivative.png|100px]]

|-

| <math>f</math> is continuous at <math>c</math> and differentiable on the immediate left and immediate right of <math>c</math> || negative || positive || strict local minimum(two-sided) || [[File:Strictlocalminwithderivativezero.png|100px]] [[File:Strictlocalminwithundefinedderivative.png|100px]]

|-

| <math>f</math> is continuous at <math>c</math> and differentiable on the immediate left and immediate right of <math>c</math> || positive || positive || neither local maximum nor local minimum, because <math>f</math> is increasing through the point|| [[File:Cubefunctionbasic.png|100px]][[File:Onethirdpower.png|100px]]

|-

| <math>f</math> is continuous at <math>c</math> and differentiable on the immediate left and immediate right of <math>c</math> || negative || negative || neither local maximum nor local minimum, because <math>f</math> is decreasing through the point|| [[File:Negativecube.png|100px]]

|}

Note that if <math>f'</math> has ambiguous sign on the immediate left or on the immediate right of <math>c</math>, the first derivative test is inconclusive.

<center>{{#widget:YouTube|id=5vlHJudKKPY}}</center>

===Relation with critical points===

The typical goal of the first derivative test is to determine whether a [[critical point]] is a point of local maximum or minimum. Hence, the test is typically applied to critical points. ''However, when applying the first derivative test, we do not need to check whether the point in question is a critical point. In other words, if the condition for being a point of local maximum or minimum is satisfied, then the point in question is automatically a critical point and this condition need not be checked separately.''

===~~Succinct ~~Short version=== At a critical point in the interior of the domain of a function where the function is continuous: * If the derivative of the function changes sign from positive (on the immediate left) to negative (on the immediate right), then the point is a point of strict local maximum.* If the derivative of the function changes sign from negative (on the immediate left) to positive (on the immediate right), then the point is a point of strict local minimum.* In general, if the derivative changes sign as we move from the immediate left of the point to the immediate right of the point, then there is a local extremum at the point. If the derivative has the same sign on the immediate left and immediate right, we ''do not'' get a local extremum at the point. <center>{{#widget:YouTube|id=8KqUt4wUtYY}}</center> ==Facts used== # [[uses::Positive derivative implies increasing]]# [[uses::Increasing on open interval and continuous at endpoint implies increasing up to and including endpoint]] ==Proof== ===Example proof of one-sided version: positive derivative on left=== All the one-sided versions have analogous proofs, so we provide a proof only for one of them. '''Given''': A function <math>f</math> and a point <math>c</math> in the domain. <math>f</math> is left continuous ''at'' <math>c</math> and differentiable on the immediate left of <math>c</math>. Further, <math>f'(x) > 0</math> on the immediate left of <math>c</math>. Explicitly, there exists <math>\delta > 0</math> such that <math>f'(x) > 0</math> for <math>x \in (c - \delta, c)</math>. '''To prove''': <math>f</math> has a strict local maximum from the left at <math>c</math>. More explicitly, we have <math>f(x) < f(c)</math> for <math>x \in (c - \delta, c)</math>.

{| class="sortable" border="1"! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation|-| 1 || <math>f</math> is increasing on the immediate left of <math>c</math>, i.e., <math>f</math> is increasing on the interval <math>(c - \delta, c)</math>. || Fact (1) || <math>f'(x) > 0</math> for <math>x \in (c - \delta, c)</math> || || given-fact direct|-| 2 || <math>f</math> is increasing from the left up to and including <math>c</math>, i.e., <math>f<~~center~~/math>~~{{#widget:YouTube~~is increasing on <math>(c - \delta,c]</math>. || Fact (2) || <math>f</math> is left continuous at <math>c</math> || Step (1) || step-given-fact direct|-| 3 ||~~id=jhRylM1xE5Y}}~~<math>f(x) < f(c)</math> for <math>x \in (c - \delta, c)</~~center~~math>|| || || Step (2) || Follows directly from Step (2).|}

'''Given''': A function <math>f</math> and a point <math>c</math> in the domain. <math>f</math> is continuous ''at'' <math>c</math> and differentiable on the immediate left and immediate right of <math>c</math>. Further, <math>f'(x) > 0</math> on the immediate left of <math>c</math> and <math>f'(x) < 0</math> on the immediate right of <math>c</math>. '''To prove''': <math>f</math> has a two-sided strict local maximum at <math>c</math>, i.e., <math>f(x) < f(c)</math> for <math>x</math> on the immediate left or the immediate right of <math>c</math>. '''Proof''': {| class="sortable" border="1"! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation|-| 1 || <math>f</math> has a strict local maximum from the left at <math>c</math> || one-sided version for strict local max from the left || <math>f</math> is continuous at <math>c</math> and <math>f'(x) > 0</math> on the immediate left of <math>c</math> || || Since <math>f</math> is continuous at <math>c</math>, it is in particular ''left'' continuous at <math>c</math>. Combining this with <math>f'(x) > 0</math> (on the immediate left) and the one-sided sign version of the first derivative test, we obtain the result.|-| 2 || <math>f</math> has a strict local maximum from the right at <math>c</math> || one-sided version for strict local max from the right || <math>f</math> is continuous at <math>c</math> and <math>f'(x) < 0</math> on the immediate right of <math>c</math> || || Since <math>f</math> is continuous at <math>c</math>, it is in particular ''right'' continuous at <math>c</math>. Combining this with <math>f'(x) < 0</math> (on the ''right'' of <math>c</math>) and the one-sided sign version of the first derivative test, we obtain the result.|-| 3 || <math>f</math> has a two-sided strict local maximum at <math>c</math> || || || Steps (1), (2) || Step-combination direct|} ==Relation with other tests== ===Other tests to determine whether critical points give local extreme values=== {| class=~~First ~~"sortable" border="1"! Test !! Quick description of how it differs from the first derivative test !! Relation with first derivative test|-| [[second derivative test]] || Instead of evaluating the sign of the first derivative on the immediate left and immediate right, we evaluate the sign of the second derivative ''at'' the point. || [[second derivative test operates via first derivative test]] (so in any situation where the second derivative test is applicable and conclusive, so is the first derivative test)<br>[[second derivative test is not stronger than first derivative test]]: There are situations where the second derivative test does not ~~require ~~apply, or is inconclusive, but the first derivative test is conclusive.|-| [[higher derivative test]]s || Instead of evaluating the sign of the first derivative on the immediate left and immediate right, we evaluate the sign of the second derivative, and if necessary, higher derivatives, ''at'' the point. || Similar to second derivative test, details need to be filled in|-| [[one-sided derivative test]] || Instead of evaluating signs of derivatives on the immediate left and immediate right of the point, we evaluate the signs of the one-sided derivatives ''at'' the point. || [[first derivative test and one-sided derivative test are incomparable]]|} ===Similar tests for functions of multiple variables=== * [[First derivative test for function of multiple variables]] ==No requirement of differentiability at the point~~=~~==

To apply the two-sided combined sign version of the first derivative test, we need ''continuity'' at the point and differentiability on the immediate left and immediate right of the point. However, we do not require differentiability ''at'' the point.

Assume that <math>f</math> is continuous at <math>c</math>, i.e., <math>\lim_{x \to c^-} f_1(x) = \lim_{x \to c^+} f_2(x) = v</math>. In that case, we can try to determine whether <math>c</math> is a point of local maximum, minimum, or neither by studying the sign of <math>f_1'</math> to the immediate left of <math>c</math> and the sign of <math>f_2'</math> to the immediate right of <math>c</math>. It is not necessary that <math>f</math> be differentiable at <math>c</math> (for more on how to differentiate piecewise functions, see [[differentiation rule for piecewise definition by interval]]).

In particular, we ''may'' be able to apply the first derivative test in these two types of situations: {| class="sortable" border=~~=Situations ~~"1"! Case !! Examples where the first derivative test works !! Pictures|-| <math>f</math> has one-sided derivatives at <math>c</math>, but these are not equal to each other || <math>f(x) := |x|</math> and <math>c = 0</math> (get local minimum) <br><math>f(x) := \left\lbrace \begin{array}{rl} x, & x < 0 \\x^2, & x \ge 0 \\\end{array}\right.</math>, <math>c = 0</math> (get neither, as the function increases through the point) || [[File:Absolutevalue.png|100px]]|-| <math>f</math> does not have well defined one-sided derivatives at <math>c</math>, but the derivative is ~~not ~~defined on the immediate left and immediate right || <math>f(x) := x^{2/3}</math>, <math>c = 0</math> (get local minimum)<br><math>f(x) := x^{1/3}</math>, <math>c = 0</math> (get neither, as the function increases through the point) || [[File:Twothirdspower.png|100px]][[File:Onethirdpower.png|100px]]|} <center>{{#widget:YouTube|id=6PJplELQB1g}}</center> ==Inconclusive and conclusivecases== ===Inconclusive cases=== Note that we consider the first derivative test to be conclusive if we can definitely conclude whether we have a local maximum, local minimum, or neither. In particular, the first derivative test is conclusive for a function that's continuous at the point, differentiable on the immediate left and immediate right of the point, and whose derivative takes constant sign (possibly allowing zero values) on the immediate left and constant sign (possibly allowing zero values) on the immediate right.

The following problems could occur when applying this test:

===~~Condition for the test to be conclusive~~Conclusive cases===

* [[First derivative test is conclusive for differentiable function at isolated critical point]]: If <math>f</math> is continuous at <math>c</math> and differentiable on the immediate left and immediate right of a [[critical point]] <math>c</math>, ''and'' <math>c</math> is an isolated critical point (i.e., there is an open interval containing it that contains no other critical points), then the first derivative test must be conclusive at <math>c</math>. In other words, we can use the first derivative test to definitively determine whether <math>c</math> is a point of local maximum, local minimum, or neither, for <math>f</math>.

* ~~In particular, the first ~~[[First derivative test is ~~always ~~conclusive for function with algebraic derivative]]: This includes polynomials and rational functions. ~~It ~~* [[First derivative test is ~~also ~~conclusive for ~~functions with piecewise definition by interval where each of the piece is a polynomial or rational ~~locally analytic function~~.~~]]