Function study, part 5: concavity and inflection points

If you are at the fifth part of this series, you have already done domain, sign, limits, asymptotes, first derivative. At this point many students breathe a sigh of relief and jump straight to the graph, considering the second derivative an optional extra. I understand: under test conditions, with time running out, it is the part that gets cut most often. But it is good to know what you really lose when you do so, and when it is instead the right choice. If you haven’t read it yet, here is the complete guide to function study with the map of all eight points.

What you really lose if you skip this step

Without the first derivative you don’t even know where the function increases or decreases: the graph you draw is, literally, wrong. Without the second derivative, on the other hand, the graph has the right shape in broad terms — it rises where it rises, falls where it falls, the maxima and minima are in the right place — but the curvature is not. A curve that rises can rise “opening upwards” like a cup, or “closing in” like a hill that flattens out at the top: your graph, without concavity, does not tell the two apart. In the marking schemes of the second maturità paper, concavity and inflection points are items in their own right, so skipping them costs specific marks, not just vagueness.

That said, under test conditions with time running out, cutting the second derivative is often the right choice if you already have domain, sign, asymptotes and first derivative well done: a graph that is qualitatively correct but without concavity is still worth the vast majority of the marks. The calculation you skip first, if you really have to choose, is this one — not the sign of $f$, not the limits.

What concavity is, first with your head and then with the formula

The second derivative $f^{\prime \prime }$ is the derivative of the first derivative: it measures how the slope changes, not the slope itself. If the slope $f^{\prime }$ is increasing as you move to the right, the graph “turns anticlockwise” and the curve looks like a cup holding water. If the slope is decreasing, the graph turns the other way and looks like a hill that would spill the water.

The analogy I always use in lessons, and the one that unlocks most students: if $f$ is the position of an object over time, $f^{\prime }$ is its velocity and $f^{\prime \prime }$ is its acceleration. A car that accelerates has increasing velocity: its position, on the position–time graph, curves “upwards”. A car that brakes has decreasing velocity: the position curves “downwards”. You are not calculating how fast the car is going (that is $f^{\prime }$), you are calculating whether it is pressing the accelerator or the brake.

Beware of the terminology: many people fall here

There is a point that creates pure confusion, and it is best to clarify it straight away: in Italy two different conventions are used and you may find both, even in the same textbook a few chapters apart.

• Some texts call the cup-shaped curve, the one with $f^{\prime \prime }>0$, “concave up” (or “concave”).
• Other texts call the same curve “convex”, reserving “concave” for the hill shape, the one with $f^{\prime \prime }<0$.

They are exactly opposite. Here I use the second convention, the more widespread one in analysis books and maturità formula sheets: convex = cup, concave up, $f^{\prime \prime }>0$; concave = hill, concave down, $f^{\prime \prime }<0$. If your book uses the other, that is perfectly fine: what matters is that you always know how to associate the name with the drawn shape and with the sign of $f^{\prime \prime }$; never trust the name alone without checking which shape it describes (in English, “concave up” and “concave down” are the unambiguous terms).

The working rule

Exactly as for increasing and decreasing with $f^{\prime }$, here you study the sign of $f^{\prime \prime }(x)$, not its numerical value:

The procedure is identical to that of the first derivative: you calculate $f^{\prime \prime }(x)$, set it greater than zero, solve the inequality, and read the sign on the intervals of the domain.

Inflection points: where concavity really changes

An inflection point is a point where the concavity changes, from convex to concave or vice versa. Here comes the most widespread mistake of the whole chapter, so I write it bluntly: $f^{\prime \prime }(x_0)=0$ is not enough to declare an inflection point. The sign of $f^{\prime \prime }$ has to change as it crosses $x_0$.

The classic counterexample is $f(x)=x^4$ at the point $x=0$. Let’s calculate: $f^{\prime }(x)=4x^3$, $f^{\prime \prime }(x)=12x^2$. At $x=0$, $f^{\prime \prime }(0)=0$. But $12x^2\ge 0$ for every $x$: the sign of $f^{\prime \prime }$ is always positive (or zero only at that isolated point), it never changes. The function stays convex to the left and right of zero. No inflection point, despite $f^{\prime \prime }(0)=0$. The graph of $x^4$ near the origin is “flatter” than a parabola, but remains a cup on both sides.

Classifying inflection points by type of tangent

At inflection points there is always a tangent line, and depending on how it behaves three cases are distinguished.

Inflection point with a horizontal tangent. It happens when, besides the sign change of $f^{\prime \prime }$, also $f^{\prime }(x_0)=0$. The tangent at the inflection point is horizontal. Canonical example: $f(x)=x^3$ at $x=0$. $f^{\prime }(x)=3x^2$, so $f^{\prime }(0)=0$; $f^{\prime \prime }(x)=6x$, which is negative for $x<0$ and positive for $x>0$: it changes sign, inflection confirmed, and with a horizontal tangent because $f^{\prime }(0)=0$.

Inflection point with an oblique tangent. It is the most common case: $f^{\prime }(x_0)\ne 0$ but $f^{\prime \prime }$ still changes sign as it crosses $x_0$. The tangent at the inflection point has any slope, different from zero.

Inflection point with a vertical tangent. It happens when $f^{\prime }(x)$ tends to $+\infty$ or to $-\infty$ (with the same sign) on both sides of $x_0$, while the function is still defined and continuous there. Example: $f(x)=\sqrt[3]{x}$ at $x=0$, with derivative

which tends to $+\infty$ both from the left and from the right of $0$ (because $x^{-2/3}$ is always positive). The tangent becomes vertical at the point, and indeed looking at the graph of the cube root you can see very well that “S” that steepens vertically near the origin.

How to write the inflectional tangent

It is a classic maturità exercise: given an inflection point at $x_0$, write the equation of the tangent line at that point. The formula is the same as for any tangent line:

Take $f(x)=x^4-6x^2$ (we will use it shortly in full) and its inflection point at $x_0=1$. We calculate $f(1)=1-6=-5$ and $f^{\prime }(x)=4x^3-12x$, so $f^{\prime }(1)=4-12=-8$. The inflectional tangent is:

Look for inflection points where $f^{\prime \prime }$ does not exist too

Another point that is often forgotten: inflection points must be looked for not only at the zeros of $f^{\prime \prime }$, but also at the points of the domain where $f^{\prime \prime }$ does not exist but $f$ does (typical with roots, absolute values, piecewise-defined functions). If the concavity changes sign there, it is still an inflection point — with a vertical, horizontal or oblique tangent depending on what $f^{\prime }$ does at that point.

The second-derivative test for maxima and minima

In part 4: monotonicity, maxima and minima we found the stationary points with $f^{\prime }(x_0)=0$ and classified them with the sign table of $f^{\prime }$. There is a shortcut, the second-derivative test, which is sometimes quicker:

• if $f^{\prime }(x_0)=0$ and $f^{\prime \prime }(x_0)>0$: $x_0$ is a relative minimum (the curve is convex, “holds water” around the stationary point);
• if $f^{\prime }(x_0)=0$ and $f^{\prime \prime }(x_0)<0$: $x_0$ is a relative maximum;
• if $f^{\prime }(x_0)=0$ and $f^{\prime \prime }(x_0)=0$: the test says nothing. You have to go back to the sign of $f^{\prime }$ before and after $x_0$ (this is exactly what happens with $x^4$ at $x=0$: $f^{\prime }(0)=0$, $f^{\prime \prime }(0)=0$, but with the sign table of $f^{\prime }$ you see straight away that it is a minimum).

It is worth using when calculating $f^{\prime \prime }$ and evaluating it at a point is quicker than building a complicated sign table of $f^{\prime }$ — typical with simple polynomial functions. It is not worth using when $f^{\prime \prime }$ is more complex to study than $f^{\prime }$ (it often happens with rational or transcendental functions), or when the test turns out inconclusive: at that point the sign table of $f^{\prime }$ remains the most reliable method, and in any case it is the one you already needed for monotonicity.

Three worked examples

Example 1 — polynomial: $f(x)=x^4-6x^2$

Let’s set $f^{\prime \prime }(x)>0$:

So the function is convex for $x<-1$ and for $x>1$, concave for $-1<x<1$. The sign changes at both points $x=-1$ and $x=1$: two inflection points. We already used them above for the tangent at $x=1$: by symmetry (the function is even, as we will see shortly) the tangent at $x=-1$ is $y=8x+3$.

Example 2 — rational function: $f(x)=\frac{x^2}{x-1}$

Differentiating again (quotient rule, with a little patience in the calculations):

Here $f^{\prime \prime }(x)$ never vanishes: for $x>1$ it is positive (convex), for $x<1$ it is negative (concave). There is no inflection point, and it is not a calculation mistake — it is that the domain is split in two by the vertical asymptote $x=1$, and the two branches have opposite concavity without ever “touching” to compare them. ”$f^{\prime \prime }$ never vanishes” is as legitimate a result as ”$f^{\prime \prime }$ vanishes at two points”: you are not looking for a mistake, you are reading what the function says.

Example 3 — exponential: $f(x)=x{e}^{-x}$

Since $e^{-x}>0$ always, the sign of $f^{\prime \prime }$ follows $x-2$: negative for $x<2$ (concave), positive for $x>2$ (convex). It changes sign at $x=2$: there is an inflection point, with an oblique tangent because $f^{\prime }(2)=e^{-2}(1-2)=-e^{-2}\ne 0$.

What concavity tells you that nothing else does

Imagine two functions that share exactly the same points of maximum, minimum and the same limits at the ends of the domain. Between two fixed points, you can join them with infinitely many different curves: some that “bulge” upwards, others that “sink”. Maxima, minima and monotonicity alone are not enough to choose which — they are compatible with both shapes. It is concavity that decides whether the graph between a maximum and a minimum passes “above” or “below” the line joining them, and it is the only piece of information in your study that establishes it. Without it, you are drawing a plausible shape, not the true one.

Symmetry: a way to save half the calculations

If $f$ is odd (symmetric about the origin, like $x^3-x$), its inflection points are symmetric about the origin: if you find an inflection point at $x_0$, you automatically have another at $-x_0$, with opposite concavity on the two corresponding sides. If $f$ is even (symmetric about the y axis, like the example $x^4-6x^2$), the concavity is symmetric about the y axis: what happens to the right of $0$ repeats mirror-wise on the left. In both cases, you calculate half the points and deduce the other half — it is exactly what saved us a second calculation in example 1.

The mistakes I see most often

1. Declaring an inflection point just because $f^{\prime \prime }(x_0)=0$. It is mistake number one of the chapter. You always need to check that the sign really changes before and after the point — keep the counterexample $x^4$ at $0$ in mind.
2. Confusing concave and convex. With two different conventions around, it happens that you swap them halfway down the page. Fix a convention at the start of the test and stay consistent, always associating the name with the drawn shape, not with the memory of the definition.
3. Forgetting the points where $f^{\prime \prime }$ does not exist. You concentrate only on the zeros of $f^{\prime \prime }$ and neglect the points (roots, absolute values) where the second derivative is not defined but there could still be a change of concavity.
4. Using the second-derivative test when it is inconclusive. If $f^{\prime \prime }(x_0)=0$, the test does not say “neither maximum nor minimum”: it says “I don’t know, go back to $f^{\prime }$”. Treating it as an answer is a conceptual mistake, not a calculation one.
5. Considering as an inflection point a point excluded from the domain where the concavity changes. Around a vertical asymptote, as in example 2, the concavity goes from one sign to the other — but that point does not belong to the domain, so it is not an inflection point. It is a mistake that comes from reading only the sign table without checking the domain.

How we work on this in lessons

• First we draw freehand two or three curves with the same monotonicity but different concavity, so the student sees what $f^{\prime \prime }$ adds before calculating a single derivative.
• We train the reflex “a zero of $f^{\prime \prime }$ is not enough, you need the sign change” on a battery of examples with and without an inflection point, including at least one $x^4$-style case.
• For rational functions, we insist on always checking the domain before declaring or excluding an inflection point, so as not to fall into the vertical asymptote mistake.
• Generally one dedicated lesson is enough to fix concavity and the classification of inflection points, if the first derivative and sign tables are already solid from a previous meeting.

With this you have everything you need on the second derivative. In part 6: how to draw the graph we put together all the pieces of the five previous parts in complete examples, from the domain to the finished graph. If you want to review the previous step, part 4: monotonicity, maxima and minima is here.

If function study remains your weak point, between domain, derivatives and graph, write to me for a free 30-minute introductory call on the contact page: we find where the reasoning gets stuck in my maths lessons and build the shortest path to get there without surprises.

The other parts of the series, if you need them out of order: part 1: domain and symmetries, part 2: intercepts and sign study, part 3: limits and asymptotes. To train on the topic in a real test, there is the Year 5 test on derivatives; the chain rule needed for $f^{\prime \prime }$ is in derivatives explained properly. The limits and asymptotes from which the graph starts, seen from the test side, are in limits, continuity and asymptotes: the Year 5 test.