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

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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 variation of first derivative test that is intended to remedy a specific defect, namely first derivative test fails for function that is discontinuous at the critical point.

Statement

Suppose f is a function and c is a point in the domain of f. 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 f at c exists Left hand limit is less than the value, i.e., \lim_{x \to c^-} f(x) < f(c) f has a strict local maximum from the left at c, regardless of the way f' behaves on the left of c
Left hand limit of f at c exists Left hand limit is greater than the value, i.e., \lim_{x \to c^-} f(x) > f(c) f has a strict local minimum from the left at c, regardless of the way f' behaves on the left of c
Left hand limit of f at c exists Left hand limit equals value, i.e., \lim_{x \to c^-} f(x) = f(c), so f is left continuous We can try to use the one-sided version of the first derivative test:
If f'(x) \ge 0 on the immediate left, then local maximum from the left
If f'(x) \le 0 on the immediate left, then local minimum from the left.
Right hand limit of f at c exists Right hand limit is less than the value, i.e., \lim_{x \to c^+} f(x) < f(c) f has a strict local maximum from the right at c, regardless of the way f' behaves on the right of c
Right hand limit of f at c exists Right hand limit is greater than the value, i.e., \lim_{x \to c^+} f(x) > f(c) f has a strict local minimum from the right at c, regardless of the way f' behaves on the right of c
Right hand limit of f at c exists Right hand limit equals value, i.e., \lim_{x \to c^+} f(x) = f(c), so f is right continuous We can try to use the one-sided version of the first derivative test:
If f'(x) \ge 0 on the immediate right, then local minimum from the right
If f'(x) \le 0 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 f at c
Either \lim_{x \to c^-} f(x) < f(c) or (f is left continuous at c and f'(x) > 0 for x on the immediate left of c) Either \lim_{x \to c^+} f(x) < f(c) or (f is right continuous at c and f'(x) < 0 for x on the immediate right of c) strict local maximum
Either \lim_{x \to c^-} f(x) > f(c) or (f is left continuous at c and f'(x) < 0 for x on the immediate left of c) Either \lim_{x \to c^+} f(x) > f(c) or (f is right continuous at c and f'(x) > 0 for x on the immediate right of c) strict local minimum
Either \lim_{x \to c^-} f(x) < f(c) or (f is left continuous at c and f'(x) > 0 for x on the immediate left of c) ither \lim_{x \to c^+} f(x) > f(c) or (f is right continuous at c and f'(x) > 0 for x on the immediate right of c) neither local maximum nor local minimum
Either \lim_{x \to c^-} f(x) > f(c) or (f is left continuous at c and f'(x) < 0 for x on the immediate left of c) Either \lim_{x \to c^+} f(x) < f(c) or (f is right continuous at c and f'(x) < 0 for x on the immediate right of c) neither local maximum nor local minimum