The quicker second-derivative test, and the case where it tells you nothing

0:00You have a critical point and you want to know what kind it is. The test you already have reads the sign of the rate on the left of it and the sign on the right, and decides from the pair. That is two readings, and each one is a walk along a stretch rather than a single substitution. There is a shorter way, and it costs one number. Take the rate of the rate at the point itself. If that number is negative, the point is a top. If it is positive, the point is a bottom.

0:34One substitution instead of two sign readings, and no walking at all. That is a real saving, and it is why the shorter rule is taught. But one number carries less than two sides do, and the whole of this video is about what gets lost. There are situations where the shorter rule returns nothing at all, and the important thing is what nothing means. Here is the rule, in three parts.

1:03First: if the rate at the point is nought and the rate of the rate is negative there, the point is a local maximum, and the value there is the local maximum value. Second: if the rate at the point is nought and the rate of the rate is positive there, the point is a local minimum. Third: if both of them are nought, the rule says nothing, and you go back to the sign test.

1:31Read the two signs again, because they are the thing people invert. Negative means a top. Positive means a bottom. And notice what the rule requires before any of that: the rate of the rate has to exist at the point. That is a hypothesis, not a formality, and it is the second thing this video measures. Before we test anything, let us get those two signs out of memory and into an argument, because a memorised sign inverts and a derived one does not.

2:07The rate of the rate is the rate at which the first rate is changing. Suppose it is negative at our point. That says the first rate is falling as we pass through. And we know one value of the first rate already: it is nought, at the point. A quantity that is falling, and that passes through nought here, was positive just before and is negative just after. Positive then negative.

2:36That is exactly the first part of the sign test, and the sign test calls that a top. So a negative rate of the rate means a top, and we did not memorise anything to get there. Run the same argument with a positive rate of the rate and the first rate is climbing through nought: negative before, positive after, which is a bottom. The shorter rule is not an independent fact. It is the sign test with the two sides worked out in advance.

3:11That is an argument, so it can be checked, and the checker behind this video runs both tests separately and compares them. The two are kept apart on purpose. The sign test reads the rate on each side of the point and never computes a second difference. The shorter test reads one number at the point and never walks a side. Neither is allowed to consult the other. Over the marked points of this video, the shorter test reaches a verdict at twelve of them.

3:44At all twelve, the two tests give the same answer. Five of those twelve have a negative rate of the rate, and the sign test calls all five a top. Seven have a positive one, and the sign test calls all seven a bottom. Nought disagreements. The signs are not a convention; they are a consequence, and they have been measured as one. Now the hypothesis, which is the part that gets skipped.

4:14The rule asks that the rate of the rate exist at the point. Only at the point, not everywhere. The obvious thing that fails is a corner, where the first rate does not exist either. Take three plus the size of the input. At nought the rate settles on minus one from the left and plus one from the right, so it settles on nothing. With no first rate there is no second one, and the shorter test cannot even begin.

4:45But the point is still a lowest value, and you can see that without any rate at all: every other value is bigger than three. So the shorter test is unavailable at a place where the answer is obvious. That is one way the hypothesis fails. It is not the only way, and the other one is much less obvious. There are points where the first rate exists perfectly well, is nought, and the second rate still refuses.

5:17Here is that second case: the input times its own size. For positive inputs it is the square. For negative inputs it is the square, turned upside down. The two halves join smoothly. The first rate at nought is nought, measured, so this is a genuine critical point and there is no corner anywhere. Now ask for the rate of the rate there. Approached from the right it settles on plus two. Approached from the left it settles on minus two.

5:50Two different numbers, so there is no rate of the rate at nought, and the shorter test has nothing to substitute. So we now have three separate routes to silence. No first rate at all, as at the corner. A first rate but no second one, as here. And both rates present with the second one equal to nought. Three routes, and the sign test answers at all three. Before the failures take over, let us see the rule doing the job it is good at, because it really is quicker when it works.

6:26Take a fourth-degree rule whose rate factors into twelve, times the input, times one less than it, times two more than it. So the rate vanishes at nought, at one, and at minus two. Three critical points. The rate of the rate is a quadratic, and we substitute into it three times. At nought it is minus twenty-four. Negative, so nought is a top, and the value there is twelve.

6:55At one it is thirty-six. Positive, so one is a bottom, and the value there is seven. At minus two it is seventy-two. Positive again, so minus two is a bottom, and the value there is minus twenty. Three critical points, three substitutions, three verdicts, and no sign walking at all. Notice that each point needed its own substitution. One evaluation classifies one point, and no more. This is the strongest case for the shorter rule, and it is a good one.

7:33There is a trap sitting right beside that example, and it is worth stopping on for a moment. The two bottoms of that rule are at one and at minus two, with values seven and minus twenty. You will meet the pair reported as minus one and minus two, with those same two values attached. Minus one is not a critical point of this rule at all. The checker measured the rate there, and it is not nought. It is positive.

8:04So the shorter test does not merely say nothing at minus one; it refuses the point outright, because the point was never a candidate. A walk across the whole stretch, told nothing about where to look, finds exactly three critical points: minus two, nought and one. The values quoted are correct. The input attached to one of them is not. The lesson is small and general: check that a point is critical before you classify it, because the shorter test will happily return a sign at a point that has no business being tested.

8:46Now the measurement this video exists for. When the rate of the rate is nought, the rule says nothing. What do people do with nothing? Very often, they treat it as an answer. They say: the test failed, so the point must be a point of inflection. That is a guess wearing the clothes of a conclusion, and it can be counted. Here is the population. Twenty-seven rules, each one built backwards from a rate that is a product of three linear factors, each factor raised to a power of one, two or three.

9:23Building them backwards matters: nothing here differentiates anything, so the rates are known by construction and the rules are assembled from them. Between them the twenty-seven rules have eighty-one critical points. Put every one of the eighty-one to the shorter test. It reaches a verdict at twenty-seven of them, and at all twenty-seven the sign test agrees. At the other fifty-four it says nothing. And of those fifty-four silent points, twenty-seven are genuine local extremes and twenty-seven are not.

10:00Exactly half. So reading silence as a point of inflection is wrong one time in two, which is the same as not reading it at all. So what do you actually do when the shorter test goes quiet? You go back to the sign test, and you finish the job. It is worth being clear about the direction of that dependence, because it is easy to get backwards. The sign test is the one that always works. It asks nothing at the point itself, which is why it reaches corners.

10:36The shorter test is a shortcut, and a shortcut has to be checked before it can be trusted. Teaching the shorter rule as the default and the sign test as an emergency measure has the dependency the wrong way round. There is one more thing worth saying about silence. It is not a sign that something has gone wrong. A repeated factor in the rate makes the rate of the rate vanish, and repeated factors are ordinary.

11:08Now the other end of the story, where the shorter rule is not just quicker but almost free. Sometimes the rate of the rate is a constant. It is the same number at every input, so you read it once and you are finished. Take two positive numbers with a fixed sum, and ask for the smallest sum of their squares. Write one number as the total less the other, and the quantity becomes a quadratic.

11:36Its rate of the rate was measured at thirty-nine different inputs, and it came back four every single time. Four is positive, so wherever the rate vanishes, that place is a bottom. One glance, and it covers the whole stretch at once. The same thing happens with a point on a span between two upright poles, choosing the point that makes the two squared distances smallest. Thirty-nine inputs, and the rate of the rate is four at all of them.

12:09And for a cylinder standing inside a cone, with the curved surface to be made as large as possible, the same measurement returns minus two everywhere. Negative, so that critical point is a top, and again one reading did the whole job. Two of those problems have answers worth stating properly, because in both cases the answer is more general than the numbers used to get it. For the two numbers with a fixed sum: the checker took nine different totals and hunted the crossing point by halving each time.

12:48Every one of the nine landed on exactly half the total, with a positive rate of the rate there. So the two numbers are equal, whatever the sum happens to be. The poles are better still. Six different spans were tried, with six different pairs of pole heights, including one pair where one pole is fifty and the other is two. Every single one put the best point at the middle of the span.

13:17The heights change the value you get. They never change where you stand. That is not remarked on where this problem is usually set, and it is the most interesting thing in it. Now the honest counterweight, because the shorter test is not always shorter. Take a four-sided figure with three of its sides equal, and make its area as large as possible. The area comes out as a linear factor multiplied by a square root.

13:49Differentiate that once and you get something workable, with the root in the denominator. Differentiate it again and the root moves further down, the algebra spreads out, and you are some way into a second page before you have a number to look at. The checker measured the rate of the rate for this one at six different inputs, and unlike the three quadratics, it is a different number at every one.

14:17That is the tell. A rate of the rate that changes from input to input is one you have to work for. The number at the critical point is negative, so the answer is right and the area is greatest there. But the sign test would have got there in two readings, and nothing warned you before you set off. Look at the first rate before you choose. If it is a polynomial, differentiate again. If it is a quotient with a root in it, read the signs instead.

14:51There is a distinction hiding in that, and it is easy to get wrong. How far a test reaches is not the same as how much work it costs. The checker took eleven critical points spread across eight different rules. Among them are two whose turning points are irrational and had to be hunted by halving, and one that is only defined on a short stretch. The shorter test returned a verdict at every one of the eleven.

15:21And at every one of the eleven, the sign test agreed with it. So on this material the shorter test is not less capable. It answers everything the sign test answers. What differs is the algebra you have to do to get the number. Choose between them on the shape of the first rate in front of you, not on which one you think is stronger. They are equally strong here. One of them is sometimes much more work.

15:52One last thing, and it is a single word. The rule's third part sends you back to decide among local maxima, local minima, and the third answer. You will often see that restated with the word local quietly dropped: maxima, minima, and the third answer. In this subject those unqualified words mean something else. They mean best across the whole stretch, not best nearby. Those are different questions, so the checker asked both.

16:25Of the marked points that the sign test calls a local top or a local bottom, there are thirteen. Only six of the thirteen are still best when the question is asked across the whole stretch. The other seven are perfectly good local answers that the whole-stretch question throws away. So dropping the word does not save space. It silently discards seven answers out of thirteen. Put it together. The shorter rule is the sign test with the two sides worked out in advance, which is why its signs can be derived and never need to be memorised.

17:08Negative is a top because a falling rate passes through nought from above. It says nothing in two situations: when the rate of the rate is nought, and when there is no rate of the rate to take. Saying nothing is not a verdict. Across eighty-one critical points it went quiet at fifty-four, and half of those were genuine extremes. When it goes quiet, go back to the sign test and finish. That test always works, which is why it is the one to learn first.

17:42And check that a point is critical before you classify it, and keep the word local where it belongs.