First Derivative Test - Precise Statement of Monotonicity Properties

Precise Statement of Monotonicity Properties

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

• If there exists a positive number r such that f is non-decreasing on (x - r, x] and non-increasing on [x, x + r), then f has a local maximum at x.
• If there exists a positive number r such that f is non-increasing on (x - r, x] and non-decreasing on [x, x + r), then f has a local minimum at x.
• If there exists a positive number r 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.
• If there exists a positive number r such that f is strictly decreasing on (x - r, x] and strictly decreasing on [x, x + r), then f is strictly decreasing on (x - r, x + r) and does not have a local maximum or minimum at x.

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.