Variation of first derivative test for discontinuous function with one-sided limits
From Calculus
Statement
Suppose is a function and is a point in the domain of . 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
Limit existence assumption | Comparison of one-sided limit and value | Conclusion |
---|---|---|
Left hand limit of at exists | Left hand limit is less than the value, i.e., | has a strict local maximum from the left at , regardless of the way behaves on the left of |
Left hand limit of at exists | Left hand limit is greater than the value, i.e., | has a strict local minimum from the left at , regardless of the way behaves on the left of |
Left hand limit of at exists | Left hand limit equals value, i.e., , so is left continuous | We can try to use the one-sided version of the first derivative test: If on the immediate left, then local maximum from the left If on the immediate left, then local minimum from the left. |
Right hand limit of at exists | Right hand limit is less than the value, i.e., | has a strict local maximum from the right at , regardless of the way behaves on the right of |
Right hand limit of at exists | Right hand limit is greater than the value, i.e., | has a strict local minimum from the right at , regardless of the way behaves on the right of |
Right hand limit of at exists | Right hand limit equals value, i.e., , so is right continuous | We can try to use the one-sided version of the first derivative test: If on the immediate right, then local minimum from the right If on the immediate right, then local maximum from the right. |
Two-sided version
We list the strict cases:
Case for left side behavior | Case for right side behavior | Conclusion for behavior of at |
---|---|---|
Either or ( is left continuous at and for on the immediate left of ) | Either or ( is right continuous at and for on the immediate right of ) | strict local maximum |
Either or ( is left continuous at and for on the immediate left of ) | Either or ( is right continuous at and for on the immediate right of ) | strict local minimum |
Either or ( is left continuous at and for on the immediate left of ) | ither or ( is right continuous at and for on the immediate right of ) | neither local maximum nor local minimum |
Either or ( is left continuous at and for on the immediate left of ) | Either or ( is right continuous at and for on the immediate right of ) | neither local maximum nor local minimum |