A sign change either side of a critical point decides which kind it is

0:00You have a critical point. A place where the rate is nought, or where there is no rate at all. What you do not have is an answer. Here are three curves. Each one has a critical point at the same place, marked the same way, and from that mark alone they are indistinguishable. On the first, the curve rises to the mark and falls away after it. A top.

0:26On the second, it falls to the mark and rises away. A bottom. On the third, it rises to the mark, goes flat for an instant, and carries straight on rising. Neither. So finding the candidate was never the answer. It was the shortlist. The question now is how to tell which of the three you are looking at, and the whole of the answer is one word: sign. Here is the rule that settles it.

0:56Take a critical point. Look at the rate just to the left of it, and at the rate just to the right of it. If the rate is positive on the left and negative on the right, the curve was climbing and then was falling. The point is a local maximum. If the rate is negative on the left and positive on the right, the curve was falling and then was climbing. The point is a local minimum.

1:26And if the sign is the same on both sides, the point is neither. That is it. One question, asked twice, once on each side. Notice what the rule never asks for. It never asks for the rate at the point. It asks for the rate near the point, on each side, and that is a different thing entirely. That distinction is not a technicality. It is the reason this test works where nothing else does.

1:56At a critical point the rate is nought, or it does not exist. Either way there is nothing there to read a sign off. So the rule steps to one side, reads a sign, steps to the other side, reads a sign, and never once evaluates anything at the point itself. Which means the point is allowed to be a corner. Take three plus the size of the input. To the left of nought the rule is three less the input, and its rate is minus one. To the right it is three plus the input, and its rate is plus one.

2:35At nought the two sides settle on different numbers, so there is no rate at nought at all. And the test does not care. Negative on the left, positive on the right: a local minimum, with value three. This checker measured every rate in this video the same way, as a limit of a difference quotient taken separately from each side. Nothing was differentiated symbolically anywhere. At the corner, the left side settles on minus one, the right on plus one, and the two together settle on nothing. Measured, not asserted.

3:15Now the third part, the one that gets read as an afterthought. It is not an afterthought. It is the part that stops you from guessing. Without it, the sentence in your head is: I have found a critical point, so it is a maximum or a minimum, and I only have to work out which. That sentence is false, and it is the commonest mistake in this whole subject.

3:41Here is how common the third case actually is. Take rates built as a product of three factors, each raised to a power of one, two or three: twenty-seven different rates, eighty-one critical points between them. Twenty-seven of those eighty-one turn out to be neither a maximum nor a minimum. Exactly a third. Not an exception. Not a special case. A third of everything the family produced. This case has a name. It is called a point of inflection, and you will meet the word spelt two ways. They mean the same thing.

4:21Whether the name is deserved is a question this video comes back to, because it turns out not always to be. Take a worked example and run the test properly. A cubic whose rate factors into three, times the input less one, times the input plus one. That rate is nought at one and at minus one, and nowhere else. Two critical points, and no opinion yet about either. Take minus one first. Just to its left the two factors are both negative, so their product is positive, and the rate is positive.

5:02Just to its right the first factor is still negative and the second has turned positive, so the product is negative. Positive then negative: a local maximum. And the value there is five. Now take one. Just to its left the product is negative. Just to its right both factors are positive, so the product is positive. Negative then positive: a local minimum. And the value there is one. Two answers per point, and both of them are asked for. Where, and how much.

5:39The point of local maxima is minus one. The local maximum value is five. Those are different sentences and you are expected to write both. That worked because of something nobody said out loud. Look at which numbers were used to read those signs. A number just below the critical point, and a number just above it. Not any old number on that side. A number close in. Here is why it matters, and it is worth a measurement rather than a warning.

6:14Take every critical point in this video, and instead of reading the sign right beside it, read the sign at every eighth of a unit out to four units on each side. That is nine hundred and forty-seven readings. A hundred and forty-eight of them disagree with the reading taken right beside the point. They fall on nine of the thirty-two sides walked, and the nearest disagreement sits only a unit and three eighths from its own point.

6:46So somewhere on this side and just beside the point are different instructions, and the difference is not small. The rule to carry is this. Pick your test number strictly between this critical point and the next one along. Between two consecutive critical points, the rate cannot change sign, because to change sign it would have to pass through nought, and that would make another critical point in between. Cross one, and you are reading somebody else's answer.

7:21Now a curve that lands on the third case. A cubic whose rate is six times the square of the input less one. That rate is nought at one. So one is a critical point, and the shortlist has one name on it. But a square is never negative. Just to the left of one the rate is positive. Just to the right of one the rate is positive. The sign never changes. The curve pauses and carries on.

7:51So one is neither a maximum nor a minimum, and a walk across the whole stretch confirms it is the only critical point there is, with nowhere the rate fails to exist. The comparison agrees. Look at the values around the point rather than at any rate, and no neighbourhood makes one a highest value or a lowest one. There is a wider fact hiding in that. A rate that never changes sign anywhere leaves a curve with no turning points at all. It climbs, or it falls, the whole way.

8:30There is a second test for this job, which uses the rate of the rate. It is often quicker, and it is not always available. It is not available at a corner, because there is no first rate there to take the rate of. So at three plus the size of the input, at a peak with a sharp point, at any join where two pieces meet at an angle, the sign test is not the shorter road. It is the only road.

9:00That is the honest reason to learn it first. But it is also sometimes just quicker, and it is worth seeing that chosen deliberately. Take the shortest distance from a point sitting up the vertical axis to a curve that squares its input. Write the distance in terms of how high up the curve you are. There is one place where its rate crosses nought. Measured at three different starting heights, that place lands half a unit below the height you started at, every time.

9:33The rate runs negative below it and positive above it, so it is a lowest value, and the distance there is the square root of one less than four times the height, halved. One sign on each side, and the question was finished. No second rate needed. Here is the one place in this subject where you can call the answer before you substitute anything. Suppose a rate arrives already factored. Something like the input less two, cubed, times the input plus one, squared, times seven times the input, less two.

10:13Three critical points: two, minus one, and two sevenths. Now think about what happens to each factor as you walk across its own root. A factor raised to an odd power changes sign there, because the thing inside it does. A factor raised to an even power does not, because a square, or a fourth power, is never negative. So the sign of the whole product changes at a root exactly when that root's exponent is odd.

10:44At minus one the exponent is two. Even. No sign change. Neither kind of extreme. At two sevenths the exponent is one, and the product runs positive then negative: a local maximum. At two the exponent is three, and the product runs negative then positive: a local minimum. All three verdicts, from parity alone. That is a prediction, so it deserves to be checked rather than believed. Take the twenty-seven rates from earlier, with their eighty-one critical points.

11:23For each one, read whether the sign actually changes, from the rate's own values on either side. Then, separately, predict whether it ought to change, from the exponent and nothing else. The two agree at all eighty-one. And both answers get used, so the agreement means something: fifty-four of the roots carry an odd exponent, and twenty-seven carry an even one. Those twenty-seven are exactly the ones that come back as neither kind of extreme. The remaining fifty-four split into thirty-four minima and twenty maxima.

12:03One more thing about how that family was built, because it matters for whether any of this is evidence. The rates were multiplied out into coefficients, and each rule was built from its rate by arithmetic on that list of coefficients. Then the rule's rate was measured back, by difference quotient, at nine inputs apiece. It matched at two hundred and forty-two of the two hundred and forty-three. The one miss is the steepest rule of the family read farthest out, where a rate above seventeen hundred is being asked to close on a millionth. The tolerance is absolute, so it bites hardest where the numbers are largest.

12:49Back to that third verdict, and to its name. A point of inflection is supposed to be a place where the curve changes the way it bends. Concave one side, convex the other. That is a claim about the curve, not a label for the cases that did not fit. So it can be checked. Bending can be measured without any rate at all. Take the value at a point, take the values a step either side, average those two, and see which is higher.

13:22If the average of the neighbours is above the middle, the curve is cupping upward there. If below, it is capping over. If they are equal, it is straight. Now run that on every point in this video that the sign test called neither a maximum nor a minimum. There are four of them. At three, the bending genuinely does change across the point. The name is earned. At the fourth, it does not.

13:51That fourth one is a corner where a gentle straight climb meets a steeper straight climb. Positive rate on the left, positive rate on the right, so the sign test says neither, and calls it a point of inflection. But both pieces are straight. The bending is nought on the left and nought on the right. Nothing changes. So the third verdict is sound as a verdict and unreliable as a name. What it really says is: not an extreme. It does not always say: the bending turned over.

14:29Now the thing that is usually left out, and it is the largest thing in this topic. Read the rule once more, slowly. If the rate is positive throughout some stretch to the left, and negative throughout some stretch to the right. Throughout. That is a hypothesis, and hypotheses can fail. What if there is no stretch on the left where the rate keeps one sign? Here is a rule where that happens. Take the square of the input, and lift it by a wave whose crowding grows without bound as the input approaches nought.

15:08Away from nought it is an ordinary smooth curve. At nought its value is nought. And every value it takes anywhere else is above nought, because the square is positive and the lift never brings it down to zero. So nought is a local minimum. Not by any rate argument. By looking at the values, which is the definition. Now ask the sign test what it thinks. Walk out from nought, a two hundred and fifty-sixth at a time, as far as a half, on each side.

15:44The rate reads on a hundred and twenty-six of the hundred and twenty-eight inputs on each side. And it does not keep one sign. It changes sign ten times on the right and nine on the left. Worse than that for the test: on each side there is a pair of inputs carrying opposite signs already within a sixty-fourth of the point. So there is no neighbourhood you could hand the test that has a single sign in it. Not a small one, not any one.

16:15Put that pair of readings to the three parts. Positive then negative? No. Negative then positive? No. The same sign on both sides? No. None of the three applies. There is a fourth answer, and the fourth answer is: this test has nothing to say here. And it is not saying nothing about nothing. There is a genuine local minimum sitting at that point, which the test has missed entirely.

16:46Take the wave away and the same square is covered at once: negative on the left, positive on the right, a local minimum. It was never the square that broke it. So hold the rule at the right strength. It is not: every critical point is a maximum, a minimum, or a point of inflection. It is: if a critical point has a definite sign on each side, then those two signs decide which of the three it is.

17:18The if is doing real work. Drop it and you have a rule that is false. In practice, for every polynomial you will meet, and every reasonable combination of the usual functions, the hypothesis holds and you will never see the fourth case. That is a fact about the functions you have been handed, not a fact about the theorem. Knowing which of the two you are relying on is most of what it means to understand a theorem at all.

17:50Now the standard exercise, run properly, because every one of these wants two answers and not one. The square: lowest at nought, and the lowest value is nought. A cubic less three times the input: highest at minus one with value two, and lowest at one with value minus two. The sine plus the cosine, on a quarter turn: the rate crosses nought at half a quarter turn, the signs run positive then negative, and the highest value there squares to two.

18:26The sine less the cosine, over a whole turn: it turns twice, once each way, at three eighths of a turn and at seven eighths. Both values square to two, and they sit on opposite sides of nought. A cubic with two turns: highest at one with value nineteen, lowest at three with value fifteen. Half the input plus two over it, on the positive numbers: lowest at two, with value two.

18:57The reciprocal of two more than the square: highest at nought, with value one half. And the input times the square root of one less it, which lives only between nought and one: its rate crosses nought at two thirds, and the highest value is twice the square root of three, over nine. That last one is the one that punishes carelessness. Walk your test inputs past one and you are asking the rule about numbers it was never defined at.

19:31So the whole of this topic is one question asked on two sides of a point. Does the sign change, and if so, which way round? Positive to negative is a top. Negative to positive is a bottom. No change is neither, and that third answer is a third of what a whole family of rates produced, not a curiosity. The rate at the point is never used, which is exactly why the test reaches corners that the other test cannot.

20:05Read your signs close in, strictly between one critical point and the next, or you will read somebody else's answer. And when a rate arrives factored, the parity of an exponent tells you the verdict before you substitute anything: odd changes sign, even does not, checked here at eighty-one roots out of eighty-one. Two answers per question, always. The point, and the value at it. And keep the if. The test speaks when each side has a sign. When neither side has one, the test is silent, and something can still be there.