Procedure for finding absolute extrema for a continuous function on a closed bounded interval
Suppose is a continuous function defined on a closed bounded interval of the form with . The extreme value theorem tells us that attains its absolute maximum value and absolute minimum value. Our goal is to determine the following four things:
- The absolute maximum value of on .
- All points in at which this absolute maximum value is attained.
- The absolute minimum value of on .
- All points in at which this absolute minimum value is attained.
Description of the procedure
Procedure via determination of local extrema
We know the following:
For a function whose domain is an interval, any point of absolute maximum must be either a point of local maximum or a point of endpoint maximum. Similarly, any point of absolute minimum must be either a point of local minimum or a point of endpoint minimum.
Therefore, the following procedure can be used to compute the absolute maximum and minimum values and all the points where these values are attained.
|Step no.||Step summary||Step details|
|1||Find points of local maximum, local minimum, endpoint maximum, endpoint minimum||First, compute all the points of local maximum for and alongside, compute all the points of local minimum . Along with this, also determine, for each of the endpoints , whether the function has a one-sided local maximum (i.e., endpoint maximum) or one-sided local minimum (i.e., endpoint minimum) at that endpoint. Use the techniques for finding local extrema to do this.|
|2||Evaluate function at all points and compare|| Evaluate the function at all the points of local maximum plus endpoint maximum. Compare these values. The largest of the values is the absolute maximum value. All the points where that largest value is attained are the points of absolute maximum.|
Evaluate the function at all the points of local minimum plus endpoint minimum. Compare these values. The smallest of the values is the absolute minimum value. All the points where that smallest value is attained are the points of absolute minimum.
The main problem with this procedure is that in Step (1), we may spend a lot of effort trying to determine the nature of local extremum using the first derivative test or second derivative test for a large number of critical points which ultimately will not turn out to be points of absolute extremum.
Procedure via determination of critical points
The idea here is to combine the observation that absolute extremum implies local extremum or endpoint extremum, along with the observation that point of local extremum implies critical point.
|Step no.||Step summary||Step details|
|1||Find all the critical points||Compute all the critical points of by computing the derivative , then determining the points in where this is either zero or undefined.|
|2||Evaluate function at critical points, endpoints, and compare||Evaluate the function at all critical points found in Step (1), and at both endpoints . Among all these values, the largest is the absolute maximum value, and all the points (among the critical points and endpoints) where this largest value is attained are the points of absolute maximum. Similarly, the smallest among these is the absolute minimum value, and all the points (among the critical points and endpoint) where this smallest value is attained are the points of absolute minimum.|
Comparison of the two procedures
The two procedures are fairly similar. There are the following key differences:
- Advantage of procedure via determination of critical points: We do not need to determine for each critical point whether it is a point of local maximum, local minimum, or neither. This saves on some effort that might have gone into using the first derivative test or second derivative test.
- Disadvantage of procedure via determination of critical points: We could potentially need to evaluate the function on a larger set of points, because there may well be many more critical points than points of local extremum. This disadvantage becomes even sharper if the goal is to only find the absolute maximum). In that case, if we first filter down to the points of local maximum and endpoint maximum, then we need only evaluate the function at those points. A similar remark applies if we need to only find the absolute minimum.
Comparing the advantage and disadvantage, we see that the main trade-off is between the ease of evaluating and comparing function values and the ease of evaluating the signs of the first derivative on intervals (if using the first derivative test) or the second derivative at critical points (if using the second derivative test). If the function is easy to evaluate and compare between points, it makes sense to simply determine the critical points and evaluate and compare values between the critical points and endpoints. If, on the other hand, the function is hard to evaluate and compare, but its derivatives are easier to study, it makes sense to use the appropriate derivative tests to find local extrema and then compare.