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

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 $c$)	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
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