PoSWW 36: Signs of Inflection

This one was brought to our attention by Simon the Likeable, and Likeable Daughter of Simon the Likeable. It is a multiple choice question from Queensland’s 2022 Methods Exam Paper 2 (tech-active) (p2), together with the subject report (p 27).

UPDATE (09/05/23)

The subject report has been updated, indicating that the correct answer is C. They have also included a footnote, indicating that an update has occurred (an openness to acknowledging error that is completely foreign to VCAA). The footnote does not indicate whether the mistake was in the grading or just in the report, but it’s a fair bet that it was the former.

UPDATE (02/06/23)

See further posts on this, here and here.

56 Replies to “PoSWW 36: Signs of Inflection”

  1. Slightly off topic, but I have to hand it to the exam writers. They truly did leave the hardest question till last, question 10 is an absolute doozy.

  2. Huh. If you solve f”(x) = 0 in the interval given you do indeed get 5 values, one of which is x = 0 . But this is not an inflection point as the curvature of the function does not change here. The VCE examiners seem to believe that f”(x) = 0 is a sufficient condition for f(x) to have an inflection point. But this is false.

    1. Yeah I was just thinking that and wondering what is wrong with me! There is no inflection point at x=0. You can generate the graph of the f(x) in GeoGebra by using a dynamic RectangleSum[] and tracing the definite integral or get an insight into the actual functon using Wolfram Alpha (pretty mind blowing result involving the Gamma function).

        1. You can also think about why an even function has no inflection point at x=0

          (Maybe I’m the only one who looks for things like that?)

    2. I’m normally not one to defend VCE examiners… but this error was from further North, up Queensland way.

      f”(x)=0 is a rather curious test for inflection points though, being neither necessary (think y=x^{\frac{1}{3}} at (0,0)) nor sufficient (think y=x^{4} at (0,0))!

  3. I don’t remember this many mistakes growing up, on exams. Especially an official exam like this, for many different students, getting a lot of prep/review. But, really, even on just regular from my teacher exams. Maybe I didn’t have eagle eye checking on things. Or maybe I’m getting false sense of prevalence since the blog highlights the bad. But still. Somehow get the impression error rate was lower back in the day.

    Wonder what is driving the problem. Obviously (a) lack of review is the most likely culprit. But I wonder if a part of the problem is (b) that this exercise was from the calculator allowed section. And (c) also that this exercise was sort of “special” in terms of looking cool/interesting and concept-y, versus a stereotypical grunt work calculus problem (like a partial fractions integration).

    Nota bene: And for that matter if you are in the “concepts more important than calculations” land, knowing to check points are really inflection points is a key concept. y=x^4 is the classic example (was told by my teacher) of a second derivative being zero, but not being an inflection point (no change in curvature, concave up on both sides of x=0).

    P.s I don’t see why a calculator is required. You can just take the second derivative and graph it (yes, without a calculator). and then see the y axis crossings (and kissing) and verify the sign changes (and one sign not changing) of f”.

    1. Of course you are correct, that one doesn’t need a calculator (although the idiotic endpoint 1.8 theoretically complicates things). Students are trained to push buttons first and think second, or never.

      Given the nature of the exams and the context of this blog, it’s not unreasonable for commenters here to chuck in a graph or whatnot.

      1. Good point, Marty, they were being jerks with the 1.8. Oops, I mean they were being thoughtful technocrats in showing how valuable the calculator is.

        It would take a little while, but is actually still technically doable on pencil/paper. Instead of trying to remember how to extract a cube root*, just do 1.8*1.8*1.8 by long multiplication and compare to the values that make cos=0. The result is ~5.8ish, which is bigger than 3pi/2 (~4.5), but less than 5pi/2 (~7+). So the crossings of f” are at x cubed = +/1 pi/2 and +/- 3pi/2.

        The kissing is at 0 (is not related to the trig, is driven by the 3xsq. You can check for sign or just realize that cos function is positive (and close to 1) near 0 angle. And of course 3xsq is nonnegative. So it is concave up on either side. So, it is a kissing, not an inflection.

        Totally, not saying I would have gotten it under exam time pressure. Just that is how I think to attack it.

        *I learned it in algebra 2 class. But even in the 80s, it was not expected knowledge in higher courses like AP Calculus. Just something to see once.

        1. “Jerks” is acceptable. And yes, you can handle the 1.8 easily enough mentally. As it happens, you also don’t need to, since the evenness of the function guarantees that there will be an even number of inflection points, which knocks out three of the four answers.

    2. We can wonder why this happens but if speculation includes (a) lack of review is the most likely culprit, then maybe its more likely to think (a) lack of competent review? Does anyone know how state exams and solutions get written and reviewed? Who are the writers and reviewers and how many are there? How are they chosen? Are they paid? Who approves the exam and the marking scheme? Do exams get marked again if the marking scheme was wrong? I dont just mean in Qld, I mean in any state such as NSW or Victoria.

      1. All good questions. I don’t know who writes/vets/etc the Victorian exams, although I have some guesses of at least some of the participants. I know a little about how, formally, the Victorian participants are chosen, but how the formal process works in practice I don’t really understand. Except, that the process clearly does not work.

        I don’t really know anything about the QLD process. I don’t watch for QLD screw-ups in the same manner, but the error above is not the first, and I would guess that they are common. A quick read suggests that the QLD exam questions are clunky in the same manner as the Vic exams. If you roll the dice enough times, …

  4. To decide whether to use a calculator may be influenced by the amount of time allowed for the question. And, if a student has a calculator on the desk for the exam, then the student might use it even for simple calculations.

    1. Yes, students and teachers should play the game. However, the students have been so indoctrinated, for years, to leap for the CAS, and so many of the exam questions are framed to be only practically doable with CAS, the overwhelming majority of students and teachers have zero ability to think properly about a mathematical question. Thank you, David. Thank you, Kaye.

    1. The answer is C. There are four inflection points.

      For the direct proof, calculate f” and note there are five points in the domain where f” = 0. Definitely x = 0 is not an inflection point (because the function is even or by explicitly looking at f”(x) near x = 0). The other four points are easily seen to be inflection points, because the trig bit of f”(x) clearly changes sign through each of these points.

      More simply (but more theoretically) for this MCQ, note that the function f is even (because f’ is odd). This means that x = 0 cannot be an inflection point and that the number of inflection points is even (they come in ± pairs). That only leaves the answer C.

      1. “and that the number of inflection points is even (they come in ± pairs).”
        Only true if the restricted domain is symmetric, [-a, a].

      2. Pushing my luck but…if the kids had just used the “when in doubt, charlie out”, they’d have got it right. 🙂

  5. It is questions like this that makes me wonder if even-odd functions have been removed or are not taught in the curriculum? Weird question design if they are in the curriculum.

    Oh and what a terrible error in the marking. DO BETTER QUEENSLAND! You’ve got a reputation as not-the-worst-state-in-math to uphold.

      1. Does it though?

        Even/odd functions have been in the VCE Methods curriculum forever but don’t seem to be assessed (much like the existence of a composite function – on which I know you have quite a strong opinion).

        If it doesn’t get assessed, teachers skim over it because there is so damn much in the course that covering everything (and finding the time for the ever-increasing SACs) is not possible in reality.

        So whilst recognizing odd/even functions and their properties as they apply to calculus by virtue of their symmetries is a really nice idea, I doubt it would be taught in any proper manner, if at all.

          1. There are a few things not much in the curriculum (aka isolated bits and pieces) that can be made more interesting, relevant and more strongly linked to the curriculum or, more practically but not less interesting, to ‘common’ exam questions.

            Strictly increasing/decreasing come to mind and the necessary and sufficient condition for all solutions to f(x) = invf(x) to be given by f(x) = x (or invf(x) = x, and use whichever equation is ‘nicer’).

            Even and odd functions are more obvious eg sketching graphs. And this point of inflection stuff is another. Knowledgeable teachers will teach all this along the way. A good SD would note these things and create more knowledgeable teachers in the process. And in the new SD maths methods now includes points of inflection. I really enjoy ‘surprising’ students with how these isolated bits and pieces can connect to the ‘mainstream’. That’s the great thing about maths.

  6. Just a couple quick notes about this question.

    1. It is possible that this is merely a colouring error on the exam report, and that QCAA graded the question correctly. For a few reasons this seems unlikely to me, but it is possible.

    2. The answer distribution is hilarious, indicating that, for whatever reason, the students had absolutely no clue how to answer the question.

    3. The choice of 1.8 endpoint is probably not responsible for point 2, but it couldn’t have helped, and it is an absurd choice.

    1. Given that the domain must be restricted so that there are a finite number of inflection points, what domain would be less absurd? Maybe [-2, 2] (but this capture another two points of inflection. Do we want this)? Positive and negative cube roots of pi/2 (but this is not pretty)? If it is a CAS-assisted exam then the questions are going to pay lip service to the machine so why is [-1.8, 1.8] any more absurd than any other choice? I’m genuinely curious. And did QCAA deliberately make the domain symmetric to exploit the even symmetry of f(x)? (Is this exploit taught in QCAA syllabus)

      I have another question. It is simple to prove that if a function is even then its derivative is odd (A => B). But this is not the result being used as a short-cut. The result being used is B => A, which is maybe less simple to prove? Does anyone know if this is taught in the QCAA syllabus?

      You’ve given QCAA a (terrific) hall pass with the (unlikely) suggestion of a coloring error.

      1. Thanks, A.

        First, you have a very good point, about what would be more natural. Yes, I was thinking something like multiples of π, but of course you are correct it would have to be multiples of roots of π. On the other hand, if a function such as \boldsymbol{\sin\left(x^3\right)}, were to naturally arise, so would roots of π as endpoints.

        Second, I would regard ±2 as more natural for exam endpoints, and makes the by-hand method more palatable. Although, I agree, including two more candidates is a drawback. It’s a reasonable conjecture that the writers first considered x = 2 and then changed it for this reason.

        Third, your “if it is a CAS-assisted exam …” argument, which I get all the damn time, is exasperating. Of course, as a practical matter, you are correct, simply because 95% of students and teachers cannot contemplate thinking if there’s a button to press as an alternative. But it’s still the reality that an intrinsically reasonable question has been significantly unreasonabled.

        Moreover, the question as written is almost a very good question to do by hand-brain, but is not largely because of the 1.8. And that, even if more hypothetical than actual, is truly insidious. It gives the message to not even contemplate thinking, because there’s a decent chance you’ll have to stop thinking, and go back to the beginning and plug in the function into the machine, all for a 90-second question.

        Fourthly, you are correct that going from f’ odd to f even is a little harder than vice versa. But what you’re really/mostly doing is going from f’ odd to f” even.

        Fifthly, i think it’s way overstating to say I’m giving a hall pass to QCAA. as I wrote, I think it’s unlikely to have been a colouring error. But, it is a possibility, and I think it is fair to raise that possibility.

  7. Was there any pushback? Why does this come a year later and in this blog? Would think a lot of people would have written QLD examiners (whoever is responsible agency). Then they would have checked, seen their mistake, and issued a clear correction/explanation.

    P.s. P.35 is interesting. (First) two of four “Additional” points apply.

    “Additional advice
    • Students should be given sufficient opportunities to consolidate their understanding of
    concavity and the relationship with the second derivative. Students need to be able to examine second derivatives to determine intervals over the domain where they are either positive or negative.
    • Students should be allowed to explore and compare the shapes of graphs of functions with the graphs of their derivatives and make connections between their shapes.”

    1. Not a year later. The subject report came out in February. I have no idea if there was public indication of the answers before that. No idea what the graders did.

            1. Yes, I guess. But it doesn’t really confirm that the question was graded incorrectly. It is still possible that the question was graded correctly but at some point “D” was incorporated in the reports.

              To be clear, I think and always thought that report error was less likely than grading error, and the second document makes it even less likely. But we don’t know.

              It’s also possible that I somehow screwed up, that the report excerpt doesn’t make the exam excerpt. But I checked many times.

              1. At least the question had a correct option. Id love to know why 3 was preferred over 2 as a distractor. 2 would seem a better distractor than 3. Speculating to the breaking point, it’s like they wanted the sole even option to be rejected (cast out from the tribe for being so obviously different to the others) and a choice made among odd options (which would imply 5 was always thought to be the correct answer). We know you hate speculation and assumption but I find it fun to wonder. What I find interesting is that QCAA only give four options rather than the traditional five in their multiple choice questions. Having five options wouldve allowed this question to propose 1, 2, 3, 4 or 5 points of inflection.

                1. I think the answers were close enough to 25-25-25-25 that there’s little point in speculating that students had any more thought than how hard to throw the dart. But I agree with your comment on having only four options: that seems weird.

  8. I’m surprised the Sauronites missed the even/odd thingie. That sort of feels like something the concept loverz would be into. If anything, I see it as a little bit of a fetish to overemphasize it, over here, with the pure math oriented instructors who love it.

    Doesn’t seem like odd/even comes into play as much as grunt multi-step algebra working does in engineering and the like. But the algebra-deprecating, concept-loving “new AP exam” emphasize it. It also fits into the classic College Board IQ test style tricky question fetish (from SATs full of double negatives and vertical angle theorems and the like).

    That said, the whole odd/even/neither is one of the steps in curve graphing/visualization. And I figure if they are pushing graphing calculators, maybe they think curve visualization is passe. I thought it was kind of cool at the time, how you’d get a general idea of what sort of beast a curve was, what with assymptotes and the like. Not needing to just go point by point like the calculator does, but having insights into the shape of the monster. Do kids still do that whole 6 step graphing functions stuff?

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