One-sided version of second derivative test
This article describes a test that can be used to determine whether a point in the domain of a function gives a point of local, endpoint, or absolute (global) maximum or minimum of the function, and/or to narrow down the possibilities for points where such maxima or minima occur.
View a complete list of such tests
This article describes a one-sided analogue of second derivative test
Contents
Statement
Suppose is a function and
is a point in the domain of
. The one-sided version of second derivative test is a slight variation of the second derivative test that helps determine, using one-sided second derivatives, whether
has a one-sided or two-sided local extremum at
.
What the test says: one-sided sign version
Continuity and differentiability assumption | Assumption on one-sided derivative at ![]() |
Assumption on one-sided second derivative at ![]() |
Conclusion about ![]() ![]() |
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strict local maximum from the left |
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strict local minimum from the left |
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inconclusive, i.e., we may have a strict local maximum, strict local minimum, or neither from the left |
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strict local maximum from the right |
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strict local minimum from the right |
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inconclusive, i.e., we may have a strict local maximum, strict local minimum, or neither from the right |
What the test says: combined sign version
Note that if either one-sided second derivative is zero, the behavior from that side, and hence the overall behavior, is inconclusive.
Continuity and differentiability assumption | Assumption on derivative | Assumption on left second derivative ![]() |
Assumption on right second derivative ![]() |
Conclusion about ![]() ![]() |
---|---|---|---|---|
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negative | negative | strict two-sided local maximum |
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positive | positive | strict two-sided local minimum |
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negative | positive | neither, it's a point of increase for the function |
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positive | negative | neither, it's a point of decrease for the function |
Facts used
Proof
One-sided version: negative second left derivative at the point
We prove just one of the four one-sided versions with nonzero one-sided second derivatives.
Given: Function and point
.
is differentiable on the immediate left of
, left differentiable at
and the left hand derivative function is itself left differentiable at
.
and
.
To prove: has a strict local maximum from the left at
.
Proof:
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | The left hand derivative ![]() ![]() ![]() |
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The definition of left hand derivative requires the function to be defined on the immediate left. | ||
2 | The left hand derivative ![]() ![]() |
Fact (1) | ![]() |
Step (1) | We apply the appropriate subcase of Fact (1) (the one-sided derivative test) to the function ![]() ![]() |
3 | The derivative ![]() ![]() ![]() |
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Steps (1), (2) | By Step (2), ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | ![]() ![]() |
Fact (2) | ![]() ![]() |
Step (3) | Follows by combining Step (3) with the appropriate one-sided case of Fact (2) (the first derivative test) and using left continuity at ![]() |
Combined sign version: piecing together the one-sided versions
The combined sign versions follow directly from the one-sided versions.