Practical:Finding local extrema

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This article provides a summary of the many approaches that can be used to find the points of local extremum for a function. The goal of the article is to describe procedures that can be used to simultaneously compute all the points of local extremum as well as points of one-sided local extremum. This includes points of local maximum, points of local minimum, and domain endpoints that are points of one-sided local maximum or minimum.

We assume for simplicity that the domain of the function is an interval or union of finitely many intervals each of which may be open, closed, or half-open half-closed.

Non-calculus versus calculus approaches

This article focuses mainly on calculus approaches to finding local extrema, so prior to beginning it, we will briefly describe the key distinction between non-calculus and calculus approaches.

Non-calculus approaches are approaches that do not involve computing the derivative of the function, but use direct comparisons of function values between points. A typical example of a non-calculus approach is the observation that x^2 attains a local minimum at x = 0 because the square of any nonzero number is positive.

First derivative test

For further information, refer: Procedure for finding local and endpoint extreme values using the first derivative test

This procedure works well for most functions, including polynomials, rational functions, other algebraic functions, and functions whose derivative is algebraic. It also works well for trigonometric functions. Note that for trigonometric functions, the set of critical points may be infinite but still discrete, as the critical points repeat periodically.

The first derivative test procedure works for finding all the local and one-sided local extrema under the following condition: the set of critical points for the function is discrete, i.e., every critical point is an isolated critical point. This follows from the fact that first derivative test is conclusive for differentiable function at isolated critical point.

The key roadblock to using the test effectively is the problem of determining the sign of the first derivative on each of the intervals between successive critical points. We recall that there are two ways of doing this:

  • If the derivative can be evaluated easily at any point, we can just pick a point in the interval and evaluate the sign of the derivative there.
  • If the derivative can be factored with the zeros of the factors corresponding to the critical points, we can perform sign analysis on each of the factors and combine. This is most useful for algebraic functions, particularly polynomials and rational functions.

If either of these approaches is feasible for the function at hand, we can use the first derivative test.

Second derivative test and higher derivative test

For further information, refer: Procedure for finding local and endpoint extreme values using the higher derivative test

In principle, this procedure is somewhat weaker than the first derivative test procedure. However, it continues to work well and be conclusive for a large class of functions including algebraic functions and most trigonometric functions.