Practical:Finding absolute extrema: Difference between revisions

Latest revision as of 17:17, 4 June 2012

General problem

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

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?
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?
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?
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:

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

Fill this in later

@@ Line 7: / Line 7: @@
 # Does <math>f</math> attain an absolute minimum value? If so, what is it? If not, is there a lower bound on the values that <math>f</math> can attain?
 # If <math>f</math> 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 <math>x^2</math> attains its absolute minimum value of 0 at <math>x = 0</math> 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==
+{{further|[[Procedure for finding absolute extrema for a continuous function on a closed bounded interval]]}}
 Suppose <math>f</math> is a [[continuous function]] defined on a closed bounded interval of the form <math>[a,b]</math> with <math>a < b</math>. The [[extreme value theorem]] tells us that <math>f</math> attains its absolute maximum value and absolute minimum value. Our goal is to determine the following four things:
@@ Line 17: / Line 27: @@
 # All points in <math>[a,b]</math> at which this absolute minimum value is attained.
-===Procedure via determination of local extrema===
+We could do this in either of two related ways:
-We know the following:
-{{quotation|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.
-{| class="sortable" border="1"
-! 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 <math>f</math>and alongside, compute all the points of local minimum <math>f</math>. Along with this, also determine, for each of the endpoints <math>a,b</math>, 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 [[Practical:Finding local extrema|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.<br>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]].
-{| class="sortable" border="1"
-! Step no. !! Step summary !! Step details
-|-
-| 1 || Find all the critical points || Compute all the critical points of <math>f</math> by computing the derivative <math>f'</math>, then determining the points in <math>(a,b)</math> where this is either zero or undefined.
-|-
-| 2 || Evaluate function at critical points, endpoints, and compare || Evaluate the function <math>f</math> at all critical points found in Step (1), ''and'' at both endpoints <math>a,b</math>. 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]].
+* Find all the local maxima, local minima, endpoint maxima, endpoint minima, then evaluate and compare to find the absolute maximum and minimum; OR
-* '''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.
+* Find all the critical points, then evaluate at critical points and endpoints and compare to find the absolute maximum and 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.
+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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