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How do you explain the concavity of a polynomial without any calculus?

As the title suggests, I am struggling to explain when given a graph of a polynomial, how we determine when it is concave up or concave down without using any calculus or tangent lines. I need to teach this because some homework questions require this knowledge and I am given the homework to go over by the coordinator of the class.

All the book does is just give examples, no reasoning nothing.

Amy B
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2132123
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  • It depends how much scientifically precise you want to be (existing answers give very precise answers in this case) or if you just want to illustrate. To illustrate, a cup facing up is concave up. – claude31 Aug 28 '20 at 08:19

6 Answers6

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Here is a proposed definition:

A function $f$ is said to be concave up on an interval $[a,b]$ if for all $x,y \in [a,b]$ with $x<y$, the line $L$ connecting $(x,f(x))$ and $(y,f(y))$ satisfies $L(t) \geq f(t)$ for all $t \in (x,y)$. It is concave down if the final inequality is reversed.

Steven Gubkin
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    In plain English (for the students), a function is concave up if, for any line segment connecting two points on the graph, the function's graph is entirely below the line segment (does not apply outside of the line segment). It is concave up "on an interval" if this is true for any two points on the graph within that interval. – Opal E Aug 27 '20 at 01:49
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    I would be cautious about using y as an independent variable in explanations to math students at this level, particularly when showing a graph - many students think y is synonymous with f(x) and values on the dependent axis, and will think you're talking about the points below the line f(x)=x, not about two different values on the independent (commonly, x) axis. – Joe Aug 27 '20 at 16:29
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    @Joe Good point! I often struggle with remembering how much students struggle with quantifier scope. Perhaps $x_1$ and $x_2$ would be better. – Steven Gubkin Aug 27 '20 at 16:52
  • Note that using $\geq$ would make lines concave up, so you probably want $>$. – Adam Aug 28 '20 at 01:43
  • Something similar can be done using support functions. – Adam Aug 28 '20 at 01:47
  • @Adam I prefer the strict inequality, because it permits piecewise linear functions as well (for instance, a line with negative slope, then horizontal line, then line with positive slope). This agrees with the Ilmari's definition of convex function as well. Under this definition, lines are the only functions which are both concave up and concave down. – Steven Gubkin Aug 28 '20 at 12:48
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    @StevenGubkin (I think you mean non-strict?) That's reasonable and agrees with notions of a convex set, which a student would probably know from convex polygons, but I'd make it a point to mention it. – Adam Aug 28 '20 at 13:07
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    As a cautionary reminder: a very large number of students at the precalculus level will find this answer's reliance on abstract symbols and rigorous mathematical formalism completely impenetrable. I hope that an educator would not use it as-is without carefully explaining each part in language the students will understand. – Tiercelet Aug 28 '20 at 14:34
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    @Tiercelet Indeed, this answer is pitched to the educator, not to the student. – Steven Gubkin Aug 28 '20 at 14:39
  • @OpalE’s comment should be an answer: it’s better than any of the current answers. – Peter LeFanu Lumsdaine Aug 29 '20 at 17:00
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As other answers have noted, a function is said to be convex (or "convex up"; I've never seen "concave up" before, although the meaning is obvious enough in context) if the line segment connecting any two points on its graph lies entirely above (or on) the graph between those points, and concave (or "convex down" / "concave down") if the line segment connecting any two points on its graph lies entirely below (or on) the graph between those points.

A rigorous algebraic definition, to complement this geometric description, is that a function $f$ is defined to be convex on a subset $S$ of its domain if and only if, for all $a,b \in S$ and all $t, s \in (0,1)$, $$t + s = 1 \implies t f(a) + s f(b) \ge f(ta + sb),$$ and concave if the opposite inequality holds (i.e. if $-f$ is convex). Further, $f$ is said to be strictly convex (or concave) if the corresponding inequality is strict.

(Note that the definition given above is often simplified by directly substituting $1 - t$ for $s$, but that somewhat obscures the underlying symmetry of the definition. The symmetric form also generalizes more readily to the various forms of Jensen's inequality.)

The connection between these two definitions is that any $x \in (a,b)$ can be written as the weighted average $x = ta + sb$, where $t + s = 1$ and both $t$ and $s$ are positive. Then $(x, f(x))$ is a point on the curve of $f$ at $x$, while $(x, y)$, where $y = t f(a) + s f(b)$, is the corresponding point on the straight line segment between the points $(a, f(a))$ and $(b, f(b))$.

Notably, this definition (in either its geometric or algebraic form) does not require the function $f$ to be differentiable or even continuous (although it can be shown that a function which is convex on an open interval must necessarily be continuous on the whole interval and differentiable at all but at most countably many points on it). Thus, it is more general than definitions based on derivatives and can be applied to more kinds of functions. For example, the function $f(x) = |x|$ is clearly convex on all of $\mathbb R$ according to this definition, even though it's not differentiable at $x = 0$. Thus, this definition, or something similar and equivalent, is usually taken as the fundamental definition of convexity, with more narrowly applicable ones like "a twice differentiable function is convex if its second derivative is non-negative" being proven as theorems.

Ilmari Karonen
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  • this definition (in either its geometric or algebraic form) does not require the function $f$ to be differentiable or even continuous --- (+1) for pointing out that convexity (like "increasing at a point" or "increasing on an interval/arbitrary-set", even or odd function, etc.) has a fairly standard meaning in which nothing about continuity or differentiability is required to state. – Dave L Renfro Aug 27 '20 at 17:30
  • For those interested, I posted a fairly detailed list of results that extend those in your last paragraph in this 28 December 2002 sci.math post. Incidentally, this was originally posted back on 14 June 2001 in the Math Forum discussion group "ap-calculus", but those posts appear to be no longer available, so it's fortunate that I copied/pasted (most of) it in that 2002 sci.math post (archived at google groups), else it would probably now be lost to me. – Dave L Renfro Aug 27 '20 at 17:33
  • Good answer. I found it easier to understand convexity with the special case t=s=0.5. By comparing the function of the average to the average of the function, so to say. A diagram with 2 or 3 examples also helps. – Eric Duminil Aug 27 '20 at 19:13
  • As a fun fact, any convex function on an interval is necessarily continuous. So convexity is a stronger condition than continuity (on an interval). – 2132123 Aug 27 '20 at 22:31
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    @2132123: Well, on an open interval, at least. On a closed interval, a convex function can be discontinuous at the endpoints. – Ilmari Karonen Aug 27 '20 at 23:45
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    It should be noted that "concave up" and "concave down" are very standard language in the US undergraduate calculus curriculum. Thomas' Calculus definitely uses it (page 204, 13th ed), and my recollection is that the terminology appears in Stewart, as well. These are, by far, the most widely used books on the topic. – Xander Henderson Aug 28 '20 at 00:37
  • This is a very good answer. I prefer my definition for precalculus students because the "connection" that you spell out in your penultimate paragraph would be very hard for students at this level. They could play with the geometrically phrased definition with pen and paper or a Desmos app. – Steven Gubkin Aug 28 '20 at 14:43
  • @XanderHenderson: Indeed, I’m familiar with concave up/down as you say, but have never heard convex up/down before, and the way they’re used in this answer seems completely counter-intuitive based on that background — one would expect convex up to mean concave down, not the same as concave up as it seems to be here. Is convex up/down really widely used with this meaning convention? – Peter LeFanu Lumsdaine Aug 29 '20 at 16:57
  • 'Concave up' and 'concave down' are standard in British secondary school education. The mnemonic rhyme 'concave up is like a cup' is a useful one for remembering the meaning. – dbmag9 Aug 29 '20 at 17:36
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You might want to discuss the etymology. There's "con", which means "with", and shows up in other words such as "converse" and "context", and "cave", which comes from "cavus", meaning "hollow", and shows up in words such as "cavity" and of course "cave". So "concave" means "with hollow". Concave down means the hollow is below the curve, and concave up means the hollow is above the curve.

Acccumulation
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One non-rigorous starting point would be that a function that could “hold water” when poured from above is concave up. (This isn’t a very robust idea and breaks down quickly on sine, for example.)

What about appealing to the apparent rate of change of the rate of increase of each function (of course this is calculus, but certain simple functions are intuitive).

Steve
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    building on the water thing, you could say that you can slosh the water around and there will never be an edge for it to flow over, it'll always be in one continuous area. – Joseph Ireland Aug 27 '20 at 12:05
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    "Do you have to build a suspension bridge over a valley or do you have to dig a tunnel through a mountain?" – JonathanZ Aug 27 '20 at 15:23
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    Even if we "slosh the water around", we can't distinguish $e^x$ from $-e^x$ this way, and we get the wrong answer for $\sqrt{|x|}$ (just as examples). – Misha Lavrov Aug 27 '20 at 16:58
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    While "holding water" (ignoring "sloshing") seems to be a nessecary condition, it is not sufficient. Sufficient would be for the water to be convex (I have no idea what a physical analogue for that would be, though). This leads to the definition of convexity using epigraphs (see e.g. https://en.wikipedia.org/wiki/Epigraph_(mathematics) ) and hypographs. (a function is "concave up" if its epigraph is convex, and "concave down" when its hypograph is convex) This will probably be a more cumbersome formalisation, but may be more intuitively appealing. – Discrete lizard Aug 28 '20 at 21:26
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I think it's helpful for students to see an important concept like this from multiple points of view, so although a definition like Ilmari Karonen's is probably the best primary definition, here is one that would also be good as calculus prep.

Suppose that for a certain point p on the graph of a polynomial, there is a unique linear function L that passes through p but doesn't cut through the graph at p. We call this a no-cut line.

A no-cut line, when defined, is also the unique tangent line, meaning intuitively that it's the best linear approximation near p.

A (nonlinear) polynomial has no-cut lines everywhere except possibly at a finite number of points, called inflection points. (Tangent lines can be defined at inflection points, but they are not no-cut lines.)

In any interval not containing inflection points, we can define the polynomial's concavity. If the slope of the no-cut line is increasing on this interval, the concavity is up, if decreasing, then down.

Remark: These definitions also carry over to many other functions, e.g., the sine and exponential. They do not work without modifications for less "well-behaved" functions such as discontinuous ones.

The notion of "cuts through" is rigorous at the level of Euclid's Proposition I, which IMO is plenty rigorous enough for a high school class.

  • I like this! And despite your advertisement to the contrary, I think the approach is rigorous and basically a topological rendition for good curves one is likely to encounter (those with reservations may use the language of neighborhoods and passage from one connected component cut out by the curve into another). – Vandermonde Aug 28 '20 at 19:02
  • @Vandermonde: Thanks :-) To see what I mean about rigor, consider the case where you use the rational-number model of the plane rather than the real-number plane. –  Aug 28 '20 at 20:47
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I would be prudent with "any 2 points" in case of by example a sinus any 2 points far enough apart will give serious problems.

to analyse a small part of the polynome:

  • choose a relevant x0, calculate y0

  • chose x1 very close to but not on x0 and calculate y1 of the polynome

  • chose x2 very close but different to x0 and x1

  • T1 = (y1 - y0)/(x1-x0) gives a proxy to the tangent between x0 and x1

  • T2 = (y2 - y1)/(x2-x1) gives a proxy to the tangent between x1 and x2

  • T2 being bigger or smaller than T1 gives a suggestion for the convexity

if T1 is > T2 then the suggested part is concave

BUT

there will be the risk that we have the bad luck of working in a zone where convexity changes ! ( example = the point x = pi of a sinus function )

the make sure that this is not the case we can work with 5 of x points instead of 3,

to make sure that T1 > T2 > T3 > T4 for concave or T1 < T2 < T3 < T4 for convex and not a mix,

if we get something like T1 > T2 > T3 < T4 then convexity has changed somewhere in the zone x2 to x4 .....

user106564
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