First Derivative Test

The first-derivative test examines a function's monotonic properties (where the function is increasing or decreasing), focusing on a particular point in its domain. If the function "switches" from increasing to decreasing at the point, then the function will achieve a highest value at that point. Similarly, if the function "switches" from decreasing to increasing at the point, then it will achieve a least value at that point. If the function fails to "switch" and remains increasing or remains decreasing, then no highest or least value is achieved.

One can examine a function's monotonicity without calculus. However, calculus is usually helpful because there are sufficient conditions that guarantee the monotonicity properties above, and these conditions apply to the vast majority of functions one would encounter.

Precise statement of monotonicity properties

Stated precisely, suppose that f is a continuous real-valued function of a real variable, defined on some open interval containing the point x.

• If there exists a positive number r > 0 such that f is weakly increasing on (x − r, x] and weakly decreasing on [x, x + r), then f has a local maximum at x. This statement also works the other way around, if x is a local maximum point, then f is weakly increasing on (x − r, x] and weakly decreasing on [x, x + r).
• If there exists a positive number r > 0 such that f is strictly increasing on (x − r, x] and strictly increasing on [x, x + r), then f is strictly increasing on (x − r, x + r) and does not have a local maximum or minimum at x.

This statement is a direct consequence of how local extrema are defined. That is, if x₀ is a local maximum point, then there exists r > 0 such that f(x) ≤ f(x₀) for x in (x₀ − r, x₀ + r), which means that f has to increase from x₀ − r to x₀ and has to decrease from x₀ to x₀ + r because f is continuous.

Note that in the first two cases, f is not required to be strictly increasing or strictly decreasing to the left or right of x, while in the last two cases, f is required to be strictly increasing or strictly decreasing. The reason is that in the definition of local maximum and minimum, the inequality is not required to be strict: e.g. every value of a constant function is considered both a local maximum and a local minimum.

Precise statement of first-derivative test

The first-derivative test depends on the "increasing–decreasing test", which is itself ultimately a consequence of the mean value theorem. It is a direct consequence of the way the derivative is defined and its connection to decrease and increase of a function locally, combined with the previous section.

Suppose f is a real-valued function of a real variable defined on some interval containing the critical point a. Further suppose that f is continuous at a and differentiable on some open interval containing a, except possibly at a itself.

• If there exists a positive number r > 0 such that for every x in (a − r, a) we have f '(x) ≥ 0, and for every x in (a, a + r) we have f '(x) ≤ 0, then f has a local maximum at a.
• If there exists a positive number r > 0 such that for every x in (a − r, a) ∪ (a, a + r) we have f '(x) > 0, then f is strictly increasing at a and has neither a local maximum nor a local minimum there.
• If none of the above conditions hold, then the test fails. (Such a condition is not vacuous; there are functions that satisfy none of the first three conditions, e.g. f(x) = x² sin(1/x)).

Again, corresponding to the comments in the section on monotonicity properties, note that in the first two cases, the inequality is not required to be strict, while in the next two, strict inequality is required.

Applications

The first-derivative test is helpful in solving optimization problems in physics, economics, and engineering. In conjunction with the extreme value theorem, it can be used to find the absolute maximum and minimum of a real-valued function defined on a closed and bounded interval. In conjunction with other information such as concavity, inflection points, and asymptotes, it can be used to sketch the graph of a function.

Resources

• Finding Relative Extrema, Khan Academy
• First Derivative Test, The Organic Chemistry Tutor

References

• Chiang, Alpha C. (1984). Fundamental Methods of Mathematical Economics (Third ed.). New York: McGraw-Hill. pp. 231–267. ISBN 0-07-010813-7.
• Marsden, Jerrold; Weinstein, Alan (1985). Calculus I (2nd ed.). New York: Springer. pp. 139–199. ISBN 0-387-90974-5.
• Shockley, James E. (1976). The Brief Calculus : with Applications in the Social Sciences (2nd ed.). New York: Holt, Rinehart & Winston. pp. 77–109. ISBN 0-03-089397-6.
• Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks Cole Cengage Learning. ISBN 978-0-495-01166-8.
• Willard, Stephen (1976). Calculus and its Applications. Boston: Prindle, Weber & Schmidt. pp. 103–145. ISBN 0-87150-203-8.