Practical:Finding absolute extrema

General problem

We are given a function $f$ of one variable, whose domain is a subset of $\mathbb {R}$. We need to answer the following questions:

1. Does $f$ attain an absolute maximum value? If so, what is it? If not, is there an upper bound on the values that $f$ can attain?
2. If $f$ does attain an absolute maximum value, what are the points of absolute maximum, i.e., at what points in the domain is the absolute maximum value attained?
3. Does $f$ attain an absolute minimum value? If so, what is it? If not, is there a lower bound on the values that $f$ can attain?
4. If $f$ does attain an absolute minimum value, what are the points of absolute minimum, i.e., at what points in the domain is the absolute minimum value attained?

Non-calculus versus calculus approaches

This article focuses mainly on calculus approaches to finding absolute 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 its absolute minimum value of 0 at $x=0$ because the square of any nonzero number is positive.

This article focuses on calculus approaches, which are generally easy to apply reliably without requiring any spark of ingenuity. However, there are some situations where calculus approaches fail due to the function being very discontinuous and/or very oscillatory. In some of these cases, the non-calculus approaches still work.

Case of a continuous function on a closed bounded interval

For further information, refer: Procedure for finding absolute extrema for a continuous function on a closed bounded interval

Suppose $f$ is a continuous function defined on a closed bounded interval of the form $[a,b]$ with $a<b$. The extreme value theorem tells us that $f$ attains its absolute maximum value and absolute minimum value. Our goal is to determine the following four things:

1. The absolute maximum value of $f$ on $[a,b]$.
2. All points in $[a,b]$ at which this absolute maximum value is attained.
3. The absolute minimum value of $f$ on $[a,b]$.
4. All points in $[a,b]$ at which this absolute minimum value is attained.

We could do this in either of two related ways:

• Find all the local maxima, local minima, endpoint maxima, endpoint minima, then evaluate and compare to find the absolute maximum and minimum; OR
• Find all the critical points, then evaluate at critical points and endpoints and compare to find the absolute maximum and minimum

For more details on the two procedures and comparison between them, see procedure for finding absolute extrema for a continuous function on a closed bounded interval.

Case of a continuous function on an open interval or an interval stretching to infinity

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