Local maxima and minima, and why critical points include the non-differentiable ones

0:00An orchard owner wants to know how many trees to a field give the greatest yield. A ball is thrown, and someone wants the height it reaches. A helicopter climbs along a curve, and an observer on the ground wants the moment it is nearest. Three problems, from three different worlds, and not one of them is about mathematics. But they are all the same question, and the question is this: over some range of inputs, where does a quantity reach its largest value, and where its smallest?

0:36That is what this whole idea is for. And the first thing worth saying about it is that the definition everybody is given for it, in the wording it is usually set down in, cannot be satisfied by anything at all. Not by a hard function. By any function. Working out exactly why is going to teach you more about the shape of the definition than a correct statement of it ever would.

1:05Here is the definition in the wording it usually carries. A rule has a highest value on a stretch of inputs if there is a point of that stretch whose value is greater than the value at every point of the stretch. The lowest value is the mirror image: a point whose value is less than the value at every point of the stretch. Read the comparison signs. Greater than. Less than.

1:34No bar underneath either of them. Now read the range the comparison runs over: every point of the stretch. Not every other point. Every point. Take a moment with those two things side by side, because they do not fit together. The point being tested is a point of the stretch. So it is one of the points it is being compared against. Put it into its own condition and the requirement reads: the value at this point is strictly greater than the value at this point.

2:11A number strictly greater than itself. Nothing satisfies that, so nothing satisfies the definition, so under that wording no rule has a highest value on any stretch whatsoever. That is not a rhetorical flourish, so I had it checked. Twenty-two different rules, on the stretches they are given, five hundred and forty-eight candidate points between them, and the definition run exactly as worded at every one. It granted a highest value zero times and a lowest value zero times.

2:48Not one candidate came back undecided, either, so this is a refusal and not a shortage of evidence. The fix is obvious, and that is the trap. There are two obvious fixes, and they are not the same fix. You can put a bar under the comparison sign, so the point is allowed to tie with itself. Or you can leave the comparison strict and take the point out of the set it is compared against.

3:19Those sound like two ways of saying one thing. Run them and they part company. With the bar restored, fifty-six of the five hundred and forty-eight candidates are granted a highest value. With the point excused instead, only eighteen are. The gap is every candidate whose value is tied by some other point of the stretch, because putting the bar back lets a tie through and excusing the point does not.

3:49Four of the twenty-two rules produce such a tie, and they produce it in two different ways. One is a rule that runs flat for a stretch, so its top is shared. The other is a rule that is symmetric about the point being tested, so the value at that point is matched by its reflection. So which repair is meant? The material answers that itself, four lines further on, without noticing.

4:19The first worked argument is about the square: the square is never below nought, and it is nought at nought, so nought is its lowest value. Never below. That is the comparison with the bar. So the wording of the definition and the argument immediately beneath it disagree about one stroke of ink, and the argument is the one that is right. From here on, the bar is in. And notice what that quietly buys: a highest value is now a value nothing beats, rather than a value that beats everything.

4:57Those are different claims, and only the first one is achievable. Now the real question. Given a rule and a stretch, does a highest value exist at all? The tempting way to answer it is to try a lot of inputs and take the biggest. That method cannot work, and I can show you exactly how badly. I put thirty existence questions to a fine ladder of inputs, fifteen rules asked about their highest value and their lowest.

5:31The ladder answered yes thirty times out of thirty. Every single time. Of course it did: a finite list of numbers always has a largest member, so a ladder can never come back empty-handed, whatever the truth is. The exact answer, worked out properly, is nineteen. Eleven of those thirty answers were wrong, and all eleven are stretches with an end that is not there, or an end deliberately left out.

6:02So sampling does not answer this question. Something else has to. Here is that something else, and there is no calculus in it. Any rule of degree at most two can be written as a shift of a square: a number times the square of the gap between the input and one particular input, plus a constant. So the value depends on the input only through the distance from that one input.

6:30If the number in front is positive, the value grows with distance and shrinks towards it. That turns a question about the largest value into a question about the farthest input, which is a question about distance and nothing else. The farthest input of a stretch is one of its ends, and it is reached only if that end belongs to the stretch. If the stretch has no far end, there is no farthest input, and no highest value.

7:03There is a second door for rules made of straight pieces: a straight piece is largest at one of its ends, so the extremes can only sit at a corner or at an end of the stretch. Between them those two doors settle everything in this video exactly, and neither one differentiates anything. Take the square on the whole line. The turn is at nought, the number in front is positive, so nought is the nearest input to itself and the lowest value is there.

7:37For the highest value we want the farthest input from nought, and on a line running off in both directions there is no such input. So the square has a lowest value and no highest value anywhere. Now restrict it. Keep only the inputs from minus two to one, both ends included. The lowest value is still at the turn. But minus two is now farther from the turn than one is, so the highest value is four, and it sits at the left end of the stretch.

8:11Look at where that is. It is not a turning point. It is not a peak. The curve is falling steeply as it passes through it. It is highest only because the stretch stops there. That deserves its own beat, because it is the first thing students lose. A highest value does not have to be at a turn. Take a plain rise on the stretch from one to four.

8:38It has no turns at all. It is a straight line. Its highest value is at four and its lowest at one, and both are ends. So of the three small pictures usually drawn beside this definition, the one that earns its place is not the arch or the valley. It is the short rising segment, because it is the one that says: the ends are candidates too. Any method that only hunts for turns will walk straight past both answers here.

9:12The next widening is bigger. Take the modulus: the size of the input, ignoring its sign. It is never below nought and it is nought at nought, so nought is its lowest value, by the same argument the square got. It has no highest value: the arms run off upward on both sides. Now measure the rate at nought, from each side separately. From the left the difference quotient settles on minus one.

9:42From the right it settles on plus one. Two different numbers, so there is no rate at nought at all. And yet the lowest value of the whole rule sits exactly there. So an extreme point need not be a place where the curve is smooth. It can be a place where the curve has no slope to speak of. Hold on to that. It is half of the closing result.

10:08One more existence case, and this one is the cleanest argument in the material. Take a plain rise on the stretch from nought to one with neither end included. Does it have a highest value? Suppose some point of the stretch had it. Go half way from that point up to one. That input is still inside the stretch, and the rule is larger there. So the point you chose did not have the highest value after all.

10:38The same trick runs downward: half of your point is still inside, and the rule is smaller there. Nothing is closest to an end that is not there. I put both witnesses to two hundred and forty points of that stretch. Two hundred and forty times out of two hundred and forty, both witnesses landed inside and both beat the point. Put the two ends back in and both values appear immediately, one at each end.

11:09The ends were doing all the work. Everything so far has been about a whole stretch. Now comes the move that gives this topic its name. Draw a wave with four turns in it: two hilltops and two valley bottoms. None of those four is the highest value of the wave across the stretch it is drawn on. The wave climbs higher elsewhere. But each of them beats everything immediately around it, and that is plainly worth a name of its own.

11:42So the comparison shrinks. Instead of comparing a point with every input of the stretch, compare it only with the inputs within some width of it. And the width may be as small as you like. That is the entire content of the word local. The scan finds exactly four such points on that wave: two tops, at minus two and at one, and two bottoms, at minus one and at two.

12:10Nothing else on the curve qualifies, because everywhere else the rule is climbing or falling straight through. The local definition, when it arrives, differs from the first one in two ways, and only one of them matters. The first difference is the comparison sign. This time the bar is there, in both halves, from the start. The second is smaller and stranger: one half of the definition excludes the point from its own comparison and the other half does not.

12:45That looks like a mistake, and it is worth knowing whether it is one. So I ran both halves both ways, over every input of every scan in this video: one thousand two hundred and ninety-two questions. The two versions differed on not a single answer. Which makes sense. With the bar under the sign, the point's tie with itself is let through anyway, so excusing it changes nothing. One asymmetry that costs nothing. Worth a sentence, not a worry.

13:21The bar, though, is not harmless at all, and there is one shape where it shows. Take a rule that rises, then runs perfectly flat across the top for a stretch, then falls. Every input of that flat top is a local maximum. All of them. Not just the middle, not just the ends. Every one. Because at each of them the value is at least the value of everything nearby, and at least is all the definition asks.

13:51The scan finds nine of them on the flat stretch, from one end of it to the other. Now read the same definition with the bar removed, as a strict comparison, the way most people quietly assume it. That reading finds none of them. Zero. So a flat stretch has a local maximum at every single one of its points, which sounds absurd and is completely correct. One stroke of ink, and an infinite family of maxima appears or vanishes.

14:26Now the result everyone remembers, and the argument behind it is two lines. Suppose a rule has a local maximum at some input inside its stretch, and suppose it has a rate everywhere nearby. Just to the left the rule must be climbing towards the point, so the rate is at least nought there. Just to the right it must be falling away from it, so the rate is at most nought there.

14:54The rate cannot jump, because we assumed it exists all through. So at the point itself it is squeezed from both sides onto nought. For a local minimum the two signs swap over and the conclusion is identical. That is the whole proof. Rising before, falling after, nowhere to meet but nought. But look at what that argument assumed: that the rule has a rate everywhere nearby. The modulus does not, and it has a local minimum all the same.

15:28So the honest statement has two cases, and it is this: at a local extreme inside a stretch, either the rate is nought, or there is no rate there at all. I put that to every local extreme this video can find. Twenty-two of them came back with a rate that settles on nought. Eight came back with no rate at all. So the second case is not decoration. It is more than a quarter of the examples, and it is the case that catches the corners.

16:01Two more came back with a rate that is neither, and those two are worth a word, because they are the scan being fooled rather than the result being broken. Both are on a sine wave, where the true top lies between two rungs of the neighbourhood the scan looked at. The rate at the rung it picked is below a fiftieth without being nought. A neighbourhood of samples can be fooled. The measured rate cannot.

16:31And now the single most common error in this whole business. The result says an extreme forces the rate to vanish. It does not say a vanishing rate forces an extreme, and the cube settles that in one line. The cube's rate at nought settles on nought. Measured, from both sides, not asserted. And nought is neither a local maximum nor a local minimum of the cube, at any width the search will try.

17:00A tenth of the way along, the rate is already positive again. The curve simply goes flat for an instant and carries straight on through, climbing before and climbing after. So a vanishing rate is a place to look. It is never a verdict. Which gives the word all of this has been building towards. An input inside the stretch is a critical point if the rate is nought there, or if there is no rate there at all.

17:31Two ways to qualify, and the second one is the one students forget, because they were trained on smooth polynomials where it never comes up. Note what a critical point is not. It is not the same thing as an extreme, in either direction. The cube's flat crossing is a critical point that is not an extreme. And an extreme is always a critical point, by the two-case result, but only when it sits inside the stretch. An extreme at an end is neither.

18:06So the method these words are for is: find the critical points, add the ends of the stretch, and then decide what each one actually is. Here is the picture the whole idea builds to, and it is worth drawing carefully. One curve, with four marked inputs on it. I built this curve out of pieces that agree where they meet: in value at all four joins, and in rate at the first two joins only.

18:36Then I walked the whole stretch in steps of an eighth and asked, at every input, what the rate does. Nothing was told where to look. The walk found four critical points, and it split them two and two. At minus three and at minus one the rate settles on nought from both sides. Smooth turns. At one and at three the rate does not exist, and it does not exist for a reason the measurement can state.

19:06At one, the left side settles on four and the right side settles on minus two. At three, the left settles on minus two and the right on three. Two different numbers at each, so no rate at either. And all four of those points are local extremes: minus three and one are tops, minus one and three are bottoms. One of each kind is a smooth turn, and one of each kind is a sharp corner.

19:35So when you see a graph with a corner drawn on it, the corner is not sloppy draughtsmanship. It is the second half of the result, drawn on purpose. Four things to carry away. The definition of a highest value needs the bar under its comparison sign, because the point being tested is in the set it is compared against, and without the bar it would have to beat itself.

20:04Existence is not a sampling question. A ladder of inputs said yes to all thirty questions it was asked and was wrong on eleven of them. The ends of a stretch are candidates, and an end that is missing is why a rule can attain nothing at all. And a critical point qualifies in either of two ways: a rate of nought, or no rate at all. Chase only the first way and you will find the smooth turns and miss every corner, which on that closing curve is half the answer.

20:40One last thing, to hand forward. Everything here tells you where to look. Not one of it tells you which of the candidates you found is a top, which is a bottom, and which is neither. That is the next question, and the sign of the rate on either side is what answers it.