First derivative test

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In calculus, the first derivative test uses the first derivative of a function to determine whether a given critical point of a function is a local maximum, a local minimum, or neither.

Intuitive explanation[edit]

The idea behind the first derivative test is to examine the monotonic properties of a function just to the left and right of a given point in its domain. If the function "switches" from increasing to decreasing at the point, then close to that point, it will achieve a highest value at that point. Similarly, if the function "switches" from decreasing to increasing at the point, then close to that point, 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.

The general idea of examining monotonicity does not depend on calculus. However, calculus is introduced 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[edit]

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

Note that in all the two cases, f is 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 required to be strict: e.g. No value of a constant function is either a local maximum or a local minimum.

Precise statement of first derivative test[edit]

The first derivative test depends on the "increasing-decreasing test", which is itself ultimately a consequence of the mean value theorem.

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 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 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 minimum at a.
  • If there exists a positive number r such that for every x in (a - r, a) (a, a + r) we have f'(x) > 0, or if there exists a positive number r such that for every x in (a - r, a) (a, a + r) we have f'(x) < 0, then f has neither a local maximum nor a local minimum at a.
  • 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.)

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 third case, strict inequality is required.

Applications[edit]

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, 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.

See also[edit]