Largest and smallest over a closed interval: candidates inside plus the two ends

0:00Here are two questions that look almost the same. What is the largest value this rule takes near the input three? And what is the largest value it takes anywhere on the stretch from one to five? The first question is local. It is answered by looking at a neighbourhood, and a rule can have several answers to it. The second is not. It asks for one number, the largest taken anywhere on the stretch, and there is only one such number.

0:32Those are different questions and they have different methods. The second one has a method so short you can learn it in a minute: make a list of inputs, work out the value at each, and take the biggest. That is genuinely the whole of it. The difficulty is not in the steps. The difficulty is in the list, because a list that is missing one input gives a confident wrong answer and looks exactly like a right one.

1:02So this video is about the list. Where it comes from, what guarantees it is finite, and the one thing that is quietly left out of the usual account of it. Start with why the question needs a stretch that carries its ends. Take the rule that adds two to its input, and ask for its largest value strictly between nought and one, ends excluded. There is no answer. Whatever input you name, there is a bigger one still inside, and it wins.

1:36You cannot show that by testing a finite list of inputs, because any finite list has a biggest member. So the checker did something else. It laid a ladder of rungs across the stretch and kept the best rung, then laid a finer ladder and kept the best rung again, six times over. With the ends left out, the six ladders gave six different inputs, and each one was beaten by the next.

2:05With the ends put back, all six ladders gave the same input, and nothing beat it. That is what closing the stretch buys. Not a nicer answer. An answer at all. Include the ends, and the search is guaranteed to finish. Leave them out, and it can run for ever. Before going further, a word about words, because this corner of the subject has too many of them. The largest value over a whole stretch gets called the absolute maximum, the global maximum, and the greatest value.

2:42The smallest gets called the absolute minimum, the global minimum, and the least value. Six words, and they name two things. Every one of the six is standard and none of them is wrong, so this is not a mistake to be corrected. It is a source of confusion to be removed. Pick one pair and use it consistently. Absolute maximum and absolute minimum is the pair that appears most often alongside this method, and it is the pair used here.

3:16What matters far more than the word is the qualifier. Local maximum and absolute maximum are different questions, and a reader who lets the two words blur will answer the wrong one. When a question says largest value on a stretch, it is the absolute one being asked for, every time. Here is a picture worth carrying around. A single unbroken curve over a closed stretch. It starts high at the left end, falls to a dip, rises to a hump, and falls again to the right end.

3:54There are two turning points inside, and they are the obvious places to look. Neither of them is the answer. The largest value on the whole stretch is at the left end, and the smallest is at the right end. The hump inside is a local maximum that is beaten, and the dip inside is a local minimum that is beaten. This is not a trick drawing. It is the ordinary case, and the measurements later in this video will say how ordinary.

4:27Carry the picture, because it is what makes the candidate list feel necessary instead of arbitrary. The list has to contain the turning points, and it has to contain the two ends, and the picture is why. Now the two results that license the method. Neither is proved here, and both are worth understanding rather than reciting. The first one is about existence. If a rule is unbroken across a stretch that carries both its ends, then it attains a largest value somewhere on that stretch, and a smallest value somewhere too.

5:08Attains is the word doing the work. Not approaches. Reaches, at some actual input you could name. That is what turns the question from a hunt into a search with a guarantee attached. It has one hypothesis that matters and you have already seen it fail: the stretch has to carry its ends. The rule that adds two to its input is as well behaved as a rule can be, and on the open stretch it attains neither.

5:39Nothing about the rule went wrong. The stretch did. So the first result says the search cannot come back empty, and that is worth knowing before you start searching. The second result narrows down where to look. Suppose the largest value over the stretch is attained at a point strictly inside it, not at either end. Then the rate there is nought. The reason is short. If the rate were positive you could step a little to the right and get a bigger value, and if it were negative you could step a little to the left. Either way the point was not the largest.

6:20The same argument runs upside down for the smallest. Now put the two results together and see what falls out. The largest value exists. It is attained somewhere. That somewhere is either one of the two ends, or a point strictly inside where the rate is nought. There are two ends, and for the rules you will meet there are finitely many inputs where the rate vanishes. So an infinite search has just become a finite list, and that is the whole trick of this topic.

6:56But read that second result once more, because it has a hypothesis in it that is easy to skim past. It assumes the rule has a rate everywhere on the stretch. If a rule has a corner in it, there is a point with no rate at all, and the result simply says nothing about that point. It does not say the point cannot be the answer. It says nothing.

7:22And a point with no rate can absolutely be the answer. Think of a V-shape: the lowest value sits exactly at the corner, where there is no rate to be nought. So the list has to be widened, and the method everyone is taught does widen it, without ever saying why. The list is: every input strictly inside where the rate is nought, every input strictly inside where the rate fails to exist, and the two ends.

7:53That second clause is the repair. The result licenses the first clause and the third, and is silent about the second. The method is right and the result behind it is narrower than the method. That is worth saying out loud rather than stepping over. Here is the method in four steps. One. Find every critical point strictly inside the stretch, meaning every input where the rate is nought or fails to exist.

8:26Two. Take the two ends. Three. Work out the value of the rule at every input on that list. Four. The largest of those values is the largest value on the stretch, and the smallest is the smallest. Two things about step four are worth noticing. It compares. It does not classify. You do not need to decide what kind of point each candidate is, and running a derivative test on each one is wasted work.

8:57And it reports two things: the value, and the input where it happens. An answer that gives only the number is half an answer. That is the method. What is left is to find out how much the two shortenings of it cost. Take a cubic on a closed stretch, and run the four steps properly. Its rate vanishes at two inputs strictly inside, and the walk in this video's checker found both of them without being told they were there.

9:29So the list has four members: those two, and the two ends. The four values are twenty-four at the left end, twenty-nine at the first turning point, twenty-eight at the second, and fifty-six at the right end. The largest value is fifty-six, at the right end. The smallest is twenty-four, at the left end. Both answers are at the ends, exactly as the picture warned. The two interior turning points are real local extremes and both of them lost.

10:03Now suppose a reader had done what most readers do, and compared only the two turning points. They would have reported twenty-nine for the largest and twenty-eight for the smallest, and been wrong twice on a single question. There is a standard example offered as the case that proves the gap, and it is worth running carefully, because it does not prove it. The rule is twelve times the input to the four-thirds power, less six times its cube root, on the stretch from minus one to one.

10:39Its rate vanishes at one eighth. Its rate fails to exist at nought, because a cube root is vertical there. So the list has four members: one eighth, nought, and the two ends. The right shape for the argument. The four values are eighteen at minus one, nought at nought, minus nine quarters at one eighth, and six at one. The largest is eighteen, at the left end. The smallest is minus nine quarters, at one eighth.

11:11Look at where those answers are. One is at an end, and the other is at the input where the rate VANISHES. The input with no rate is a candidate that loses. So the shortened list, the one that cannot carry a point with no rate, still gets both answers right here. This example has the right shape and does not bite. So where does it bite? That is a question with a number for an answer, and the checker went and got it.

11:44Sixty-eight rules on one closed stretch. Four base polynomials, each taken plain and each taken with a corner deliberately put at one of four inputs inside, leaning one of four ways. Each rule asked for its largest and its smallest value. A hundred and thirty-six answers. The truth for every one of them was found by comparison alone. A ladder across the stretch, refined around its best rung, taking no rate anywhere.

12:17Then three candidate lists were scored against that truth. The full list, with both kinds of critical point and the two ends, was right at all one hundred and thirty-six. The shortened list, the one that cannot carry a point with no rate, was right at a hundred and eight. It lost twenty-eight. And those twenty-eight are exactly the answers that sit on a corner. Not most of them. The same set, with nothing in either that is not in the other.

12:49That is the gap, measured. It costs nothing until the answer is at a corner, and then it costs the answer. Now the other shortening, which is the one people actually make. Forget the ends. Compare the turning points and take the best. On the same hundred and thirty-six answers, that list was right at sixty-seven. It is wrong about half the time, and the reason is not subtle: seventy-one of the hundred and thirty-six answers sit at an end of the stretch, where a list with no ends in it cannot reach them.

13:28Put those two numbers beside each other. The gap in the theorem costs twenty-eight answers out of a hundred and thirty-six. Forgetting the ends costs sixty-nine. The subtle omission is real and it is worth knowing. The ordinary one is more than twice as expensive. And the ends are the easy half of the list. They are handed to you in the question. There is one more thing in those numbers, and it is the sharpest thing in this video.

14:01Seventy-one answers sit at an end. The turning-points list should therefore be wrong seventy-one times. It is wrong sixty-nine times. At the other two, it returns the right value. Here is one of them. A cubic on the stretch from minus two to two. Its smallest value is minus two, attained at the left end. It also has a turning point inside, at one, and the value there is also minus two.

14:33So a reader who compared only the turning points reports minus two, and minus two is correct. They have the right number attached to the wrong input, and nothing in their working tells them so. Scored on the value alone that list gets sixty-seven. Scored on the value and the input, which is what the question asks for, it gets sixty-five. Two answers that look right and are not. That is why step four reports both things.

15:06Sometimes there are no ends to add, and then the method is missing its last step. Here is the shape of it. A point runs along a curve and you want the one nearest a fixed point somewhere off it. Two useful moves before any calculus. Minimise the square of the distance rather than the distance, because the square root only rises and so the two are smallest at the same input.

15:34And notice there is no stretch. The point can be anywhere along the curve. The squared distance has exactly one input where its rate vanishes, and the checker found it by walking and then halving: the input one. There are no ends. So the candidate list has a single member, and step four has nothing to compare it against. The list alone cannot tell you whether that one candidate is the largest or the smallest or neither.

16:06So something has to be added, and the addition is one line: evaluate somewhere else and compare. The squared distance at the candidate is five. At the input nought it is nine. Five is the smaller, so the candidate is a minimum, and the nearest distance is the square root of five. That extra line is not decoration. Without it the answer is not established, and it is the line most often left out.

16:38The checker worked a set of these items rather than reading any answer off, and a few of them are worth putting up. Six stretches, each asked for both answers with the input reported alongside the value. Twelve answers, and eight of the twelve sit at an end. The cube on a symmetric stretch has one turning point inside and it wins neither question. Both answers are at the ends: eight and minus eight.

17:09One item has its turning point exactly AT the right end. There is nothing strictly inside at all, and the ends carry the whole answer. Two items have answers that are not whole numbers, and the turning point was hunted by halving rather than solved for. The sine plus the cosine over a half turn peaks at the square root of two. One item asks for the parameter rather than the answer: it fixes where the largest value is to be, and asks what the parameter must be.

17:44Run at three values, one below, one above, and one between, the rate at the stated input reads negative, then positive, and nought in between, and only in the middle case does the answer land exactly on the stated input. One item asks for the inputs rather than the value. The sine of twice the input reaches its largest value, one, at two different inputs over a whole turn, and both are wanted.

18:16And one item is not a calculus question at all. Written in terms of the sine it becomes a downward quadratic in a quantity between nought and one, and it peaks at five quarters in a single line. Look at the rule before differentiating it. Sometimes there is nothing to differentiate. What it comes to. The largest and smallest values over a closed stretch are found by listing, evaluating and comparing, and the whole difficulty is the list.

18:52The list is the critical points strictly inside, meaning where the rate is nought OR fails to exist, together with the two ends. Two results license it. One says the search cannot come back empty, and it needs the ends. The other says an answer strictly inside must have a nought rate, and it needs the rule to have a rate at all. The second is narrower than the method it licenses, and over a hundred and thirty-six measured answers that gap costs twenty-eight.

19:27Leaving the ends off costs sixty-nine, and twice over it costs them invisibly, by returning the right number from the wrong input. Report the value and the input. Check whether there are ends. And do not classify anything you were not asked to classify.