Difference between revisions of "First derivative test"
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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. | 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. |
Revision as of 19:58, 3 May 2012
Contents
Statement
What the test is for
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
Suppose is a function defined at a point
.
Then, we have the following:
Continuity and differentiability assumption | Hypothesis on sign of derivative | Conclusion |
---|---|---|
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What the test says: combined sign versions
Suppose is a function defined around a point
(i.e.,
is defined in an open interval containing
) and is continuous at
. We do not care whether
is differentiable at
; however, the test makes sense only if
is differentiable on the immediate left and immediate right of
.
Then, we have the following (we list only the strict cases in the table below):
Continuity and differentiability assumption | Sign of the derivative ![]() ![]() |
Sign of ![]() ![]() |
Conclusion for ![]() ![]() |
---|---|---|---|
![]() ![]() ![]() |
positive | negative | strict local maximum (two-sided) |
![]() ![]() ![]() |
negative | positive | strict local minimum (two-sided) |
![]() ![]() ![]() |
positive | positive | neither local maximum nor local minimum, because ![]() |
![]() ![]() ![]() |
negative | negative | neither local maximum nor local minimum, because ![]() |
If we replace positive by nonnegative and negative by nonpositive in the rows corresponding to strict local maximum and strict local minimum, we could potentially lose the strictness.
Note that if has ambiguous sign on the immediate left or on the immediate right of
, the first derivative test is inconclusive.
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.
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.
Facts used
- Positive derivative implies increasing
- 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 and a point
in the domain.
is left continuous at
and differentiable on the immediate left of
. Further,
on the immediate left of
. Explicitly, there exists
such that
for
.
To prove: has a strict local maximum from the left at
. More explicitly, we have
for
.
Proof:
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | ![]() ![]() ![]() ![]() |
Fact (1) | ![]() ![]() |
given-fact direct | |
2 | ![]() ![]() ![]() ![]() |
Fact (2) | ![]() ![]() |
Step (1) | step-given-fact direct |
3 | ![]() ![]() |
Step (2) | Follows directly from Step (2). |
Alternate version of proof: Instead of using Facts (1) and (2) in separate steps, we can use the version of Fact (1) for the one-sided closed interval , using continuity at
and the positive sign of the derivative both together. Conceptually, this is the same proof, but the presentation differs somewhat.
Example proof of combined sign version: strict local maximum
We give the proof for the strict local maximum case. Other cases are analogous.
Given: A function and a point
in the domain.
is continuous at
and differentiable on the immediate left and immediate right of
. Further,
on the immediate left of
and
on the immediate right of
.
To prove: has a two-sided strict local maximum at
, i.e.,
for
on the immediate left or the immediate right of
.
Proof:
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | ![]() ![]() |
one-sided version for strict local max from the left | ![]() ![]() ![]() ![]() |
Since ![]() ![]() ![]() ![]() | |
2 | ![]() ![]() |
one-sided version for strict local max from the right | ![]() ![]() ![]() ![]() |
Since ![]() ![]() ![]() ![]() ![]() | |
3 | ![]() ![]() |
Steps (1), (2) | Step-combination direct |
Relation with other tests
Other tests to determine whether critical points give local extreme values
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) second derivative test is not stronger than first derivative test: There are situations where the second derivative test does not apply, or is inconclusive, but the first derivative test is conclusive. |
higher derivative tests | 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
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.
Thus, for instance, the first derivative test can be used to study the behavior of a function with a piecewise definition by interval, such that the function is changing definition at the point. Explicitly, it can be used to study functions of the form:
Assume that is continuous at
, i.e.,
. In that case, we can try to determine whether
is a point of local maximum, minimum, or neither by studying the sign of
to the immediate left of
and the sign of
to the immediate right of
. It is not necessary that
be differentiable at
(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:
Case | Examples where the first derivative test works |
---|---|
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Relation between first derivative test and second derivative test
For further information, refer: Second derivative test operates via first derivative test, second derivative test is not stronger than first derivative test
The first derivative test is strictly more powerful than the second derivative test. In other words, in any situation where the second derivative test is applicable and conclusive, the first derivative test is also applicable and conclusive. However, there are many situations where the first derivative test is conclusive but the second derivative test is not.
Relation between first derivative test and direct analysis of one-sided derivatives
For further information, refer: first derivative test and one-sided derivative test are incomparable
The first derivative test and the one-sided derivative test are incomparable: there are situations where one test is conclusive and the other isn't, and vice versa.
When is the test conclusive and inconclusive?
Situations when the test is inconclusive
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:
What problem do we run into? | What kind of trouble can we have? | Link to example | Can this be fixed? | Picture |
---|---|---|---|---|
The function is not continuous at the critical point | We may be able to do sign analysis of the derivative on the immediate left and immediate right, but draw incorrect conclusions by applying the one-sided or combined sign version of the first derivative test. A priori, all the possibilities (local maximum, local minimum, neither) remain open. | first derivative test fails for function that is discontinuous at the critical point | If the function has one-sided limits at the critical point: variation of first derivative test for discontinuous function with one-sided limits | |
The function is not differentiable at points on the immediate left and/or immediate right of the point | We will not be able to make a meaningful statement about the sign of the derivative on the immediate left and/or immediate right. Thus, it will not be possible to apply the first derivative test. All the possibilities (local maximum, local minimum, neither) remain open. | first derivative test fails for function that is not differentiable near critical point | Not directly. We have to use other methods. | |
The derivative of the function has oscillatory (ambiguous) sign on the immediate left and/or immediate right of the point | We cannot do sign analysis on the derivative on the immediate left and/or immediate right. Thus, it will not be possible to apply the first derivative test. All the possibilities (local maximum, local minimum, neither) remain open. | First derivative test is inconclusive for function whose derivative has ambiguous sign around the point | ![]() |
Condition for the test to be conclusive
- First derivative test is conclusive for differentiable function at isolated critical point: If
is continuous at
and differentiable on the immediate left and immediate right of a critical point
, and
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
. In other words, we can use the first derivative test to definitively determine whether
is a point of local maximum, local minimum, or neither, for
.
- In particular, the first derivative test is always conclusive for polynomials and rational functions. It is also conclusive for functions with piecewise definition by interval where each of the piece is a polynomial or rational function.