View source for Variation of first derivative test for discontinuous function with one-sided limits

{{maxmin test}}
{{remedy variation|
original = first derivative test|
defect = first derivative test fails for function that is discontinuous at the critical point}}
==Statement==

Suppose <math>f</math> is a function and <math>c</math> is a point in the domain of <math>f</math>. This statement is a variation of the [[first derivative test]] that helps deal with situations where a function has one-sided limits but is not necessarily continuous at the [[critical point]] of interest. It can be viewed as a remedy for the fact that the [[first derivative test fails for function that is discontinuous at the critical point]].

===One-sided version===

{| class="sortable" border="1"
! Limit existence assumption !! Comparison of one-sided limit and value!! Conclusion
|-
| Left hand limit of <math>f</math> at <math>c</math> exists || Left-hand limit is less than the value, i.e., <math>\lim_{x \to c^-} f(x) < f(c)</math> || <math>f</math> has a strict local maximum from the left at <math>c</math>, ''regardless'' of the way <math>f'</math> behaves on the left of <math>c</math>
|-
| Left hand limit of <math>f</math> at <math>c</math> exists || Left-hand limit is greater than the value, i.e., <math>\lim_{x \to c^-} f(x) > f(c)</math> || <math>f</math> has a strict local minimum from the left at <math>c</math>, ''regardless'' of the way <math>f'</math> behaves on the left of <math>c</math>
|-
| Left hand limit of <math>f</math> at <math>c</math> exists || Left-hand limit equals value, i.e., <math>\lim_{x \to c^-} f(x) = f(c)</math>, so <math>f</math> is left continuous || We can try to use the one-sided version of the [[first derivative test]]:<br>If <math>f'(x) \ge 0</math> on the immediate left, then local maximum from the left<br>If <math>f'(x) \le 0</math> on the immediate left, then local minimum from the left.
|-
| Right hand limit of <math>f</math> at <math>c</math> exists || Right-hand limit is less than the value, i.e., <math>\lim_{x \to c^+} f(x) < f(c)</math> || <math>f</math> has a strict local maximum from the right at <math>c</math>, ''regardless'' of the way <math>f'</math> behaves on the right of <math>c</math>
|-
| Right hand limit of <math>f</math> at <math>c</math> exists || Right-hand limit is greater than the value, i.e., <math>\lim_{x \to c^+} f(x) > f(c)</math> || <math>f</math> has a strict local minimum from the right at <math>c</math>, ''regardless'' of the way <math>f'</math> behaves on the right of <math>c</math>
|-
| Right hand limit of <math>f</math> at <math>c</math> exists || Right-hand limit equals value, i.e., <math>\lim_{x \to c^+} f(x) = f(c)</math>, so <math>f</math> is right continuous || We can try to use the one-sided version of the [[first derivative test]]:<br>If <math>f'(x) \ge 0</math> on the immediate right, then local minimum from the right<br>If <math>f'(x) \le 0</math> on the immediate right, then local maximum from the right.
|}

===Two-sided version===

We list the strict cases:

{| class="sortable" border="1"
! Case for left side behavior !! Case for right side behavior !! Conclusion for behavior of <math>f</math> at <math>c</math>
|-
| Either <math>\lim_{x \to c^-} f(x) < f(c)</math> or (<math>f</math> is left continuous at <math>c</math> and <math>f'(x) > 0</math> for <math>x</math> on the immediate left of <math>c</math>) || Either <math>\lim_{x \to c^+} f(x) < f(c)</math> or (<math>f</math> is right continuous at <math>c</math> and <math>f'(x) < 0</math> for <math>x</math> on the immediate right of <math>c</math>) || strict local maximum
|-
| Either <math>\lim_{x \to c^-} f(x) > f(c)</math> or (<math>f</math> is left continuous at <math>c</math> and <math>f'(x) < 0</math> for <math>x</math> on the immediate left of <math>c</math>) || Either <math>\lim_{x \to c^+} f(x) > f(c)</math> or (<math>f</math> is right continuous at <math>c</math> and <math>f'(x) > 0</math> for <math>x</math> on the immediate right of <math>c</math>) || strict local minimum
|-
| Either <math>\lim_{x \to c^-} f(x) < f(c)</math> or (<math>f</math> is left continuous at <math>c</math> and <math>f'(x) > 0</math> for <math>x</math> on the immediate left of <math>c</math>) || Either <math>\lim_{x \to c^+} f(x) > f(c)</math> or (<math>f</math> is right continuous at <math>c</math> and <math>f'(x) > 0</math> for <math>x</math> on the immediate right of <math>c</math>) || neither local maximum nor local minimum
|-
| Either <math>\lim_{x \to c^-} f(x) > f(c)</math> or (<math>f</math> is left continuous at <math>c</math> and <math>f'(x) < 0</math> for <math>x</math> on the immediate left of <math>c</math>) ||   Either <math>\lim_{x \to c^+} f(x) < f(c)</math> or (<math>f</math> is right continuous at <math>c</math> and <math>f'(x) < 0</math> for <math>x</math> on the immediate right of <math>c</math>) || neither local maximum nor local minimum
|}