Maxima, minima and monotonicity

There is a precise moment at which I see most of my students get stuck in this chapter: they have calculated $f^{'} (x)$ correctly, perhaps with a long calculation too, and then they stay there staring at it as if it should tell them something by itself. They try substituting numbers, look for a pattern, wait for an inspiration. But the first derivative is not there to be looked at: it is there to be studied for its sign, exactly as you would with any other inequality. Once you understand this, the chapter unlocks.

This is the fourth piece of the series on function study. If you haven’t done so yet, start from the complete guide to function study to have the map of all the steps, and catch up on part 3: limits and asymptotes if you haven’t yet closed that piece — here I take for granted that domain, symmetries and asymptotes are already in your pocket. I don’t redo the differentiation rules here: you will find those in derivatives explained properly. Here the derivative is a tool, not the topic.

The sign, not the value

Let’s say it again because it is the concept that holds up the whole chapter: of $f^{'} (x)$ you don’t care how much it is worth at a point, you care whether on a certain interval it is positive, negative or zero. The numerical value of the derivative (the exact slope of the tangent) is only needed in specific problems — finding the equation of the tangent, an optimisation problem with a numerical constraint. For function study, $f^{'} (x) = 3$ and $f^{'} (x) = 300$ tell exactly the same story: the function is rising. All you need is the sign.

Concretely, the procedure is: you calculate $f^{'} (x)$ , write it as a fraction or product of factors, and study $f^{'} (x) > 0$ like any other inequality — with all the tools you already use for rational inequalities, with radicals, with exponentials.

The link between the sign of f’ and monotonicity

The theorem you use is simple to state:

f^{'} (x) > 0 on an interval ⟹ f is increasing on that interval

f^{'} (x) < 0 on an interval ⟹ f is decreasing on that interval

The word that counts in this sentence is interval. Increasing and decreasing are properties that hold on a continuous piece of the domain, not on “the whole domain” taken generically. If the domain is split — typically by a vertical asymptote, as in rational functions — you cannot say “the function is increasing on the whole domain” even if the sign of $f^{'}$ is the same to the left and to the right of the excluded point. You must list the intervals separately: increasing on $(- \infty, 1)$ and increasing on $(1, + \infty)$ are two different statements from “increasing on $R ∖ {1}$ ”, because at that point the function is not even defined and the graph has a jump (the asymptote). It is one of the most common mistakes I see written in tables, and I flag it already now because you will meet it again in the worked examples below.

The domain of f’ can be smaller than the domain of f

Before studying the sign, always check the domain of the derivative. It is not guaranteed to coincide with that of $f$ : it can be smaller.

Classic example:

f (x) = x, domain [0, + \infty)

The derivative is

f^{'} (x) = \frac{1}{2 x}

whose domain is $(0, + \infty)$ : zero is excluded, because the denominator vanishes there. This is not a technical detail to ignore — it means that at $x = 0$ the function is defined but not differentiable, and as you will see shortly that point still has to be studied separately, because it can hide a maximum or a minimum. Skip this check and you risk losing an important point of the function without even noticing.

Stationary points: candidates, not certainties

Stationary points are the points where $f^{'} (x) = 0$ : there the tangent to the graph is horizontal. Be careful how you name them mentally — they are not “the maxima and minima”, they are the candidates to be so. Finding $f^{'} (x_{0}) = 0$ tells you only that at $x_{0}$ the function has a flat tangent: it is up to you to establish what happens around that point.

First-derivative sign test

It is the test I use almost always, because it is the one that also gives you the monotonicity, so you do it anyway. You look at how the sign of $f^{'}$ changes when crossing $x_{0}$ :

from $+$ to $-$ : the function stops increasing and starts decreasing → relative maximum
from $-$ to $+$ : the function stops decreasing and starts increasing → relative minimum
sign unchanged (for example $+$ before and $+$ after): the function keeps increasing (or decreasing) as if nothing had happened → it is neither a maximum nor a minimum, it is an inflection point with a horizontal tangent

The textbook example for the third case is $f (x) = x^{3}$ at $x_{0} = 0$ . Here $f^{'} (x) = 3 x^{2}$ , which is $\geq 0$ everywhere and vanishes only at $0$ . The sign never changes (it is always $\geq 0$ ), so $f$ is always increasing, even crossing $x = 0$ : that point is an inflection point with a horizontal tangent, not a minimum, even though the tangent there is flat.

Second-derivative test (a preview)

There is an alternative when calculating the sign of $f^{'}$ by hand is inconvenient: you look at the sign of $f^{''} (x_{0})$ .

$f^{'} (x_{0}) = 0$ and $f^{''} (x_{0}) > 0$ → relative minimum (the function is concave up there)
$f^{'} (x_{0}) = 0$ and $f^{''} (x_{0}) < 0$ → relative maximum (concave down)
$f^{'} (x_{0}) = 0$ and $f^{''} (x_{0}) = 0$ → the test concludes nothing, you have to go back to the sign of $f^{'}$

This test is convenient but has a clear limit: if $f^{''} (x_{0}) = 0$ you have no answer, and you still have to go back to the sign method. I discuss it at length, with concavity and real inflection points, in part 5: concavity and inflection points. For now keep it as a backup tool: the sign of $f^{'}$ remains the method that always works.

Maxima and minima: relative versus absolute

Here a distinction is at stake that very many students treat as synonyms, and they are not.

A relative maximum (or minimum) is a point that is the highest (or lowest) only in its neighbourhood: looking a little to the right and a little to the left, at no other nearby point does the function exceed that value. An absolute maximum (or minimum) is the highest (or lowest) value over the whole domain (or over the set you are considering).

The delicate point: a relative minimum can lie higher than a relative maximum, if they are on different branches of the graph. It is not an oddity, it is the norm in functions with vertical asymptotes. Take the function we study in full below:

f (x) = \frac{x ^{2}}{x - 1}

It has a relative maximum at $x = 0$ worth $f (0) = 0$ , and a relative minimum at $x = 2$ worth $f (2) = 4$ . The “minimum” is worth more than the “maximum”, because they are on two branches separated by the asymptote at $x = 1$ and there is no contradiction: the name relative maximum/minimum describes the local behaviour, not a global comparison.

To find the absolute maxima and minima on a domain, you compare:

the values of $f$ at the stationary points and at the points of non-differentiability inside the domain;
the values of $f$ at the ends of the domain, if they are included (closed domain);
the limits of $f$ at the ends, if they are excluded (open domain, or at infinity) — in that case the absolute maximum or minimum might not exist at all, because you approach a value without ever reaching it.

A theoretical reference in one line, because it often comes back in constrained problems: Weierstrass’s theorem guarantees that if $f$ is continuous on a closed and bounded interval $[a, b]$ , the absolute maximum and minimum always exist, even if you perhaps don’t find them among the stationary points (they might fall right at the ends $a$ and $b$ ).

Points of non-differentiability: where marks are won or lost in the test

This is the part that really separates a well-set-up test from a very-well-set-up one. A point where $f$ is continuous but not differentiable can still be a maximum or a minimum, and if you forget it in the sign table you lose it. There are three cases, and they are told apart by looking at the limits of the one-sided derivatives (equivalently, the limit of the difference quotient from the right and from the left).

Corner point: the one-sided derivatives exist, are finite, but are different from each other. The graph has a sharp edge. Example: $f (x) = ∣ x ∣$ at $x = 0$ :

f^{'} (0^{+}) = 1, f^{'} (0^{-}) = - 1

Finite, different. At that point there is, not by chance, a minimum (relative and absolute). If the one-sided derivatives remain an abstract formula, in the interactive figure on the derivative choose $∣ x ∣$ and look at the secants: those from the right all sit at slope 1, those from the left at −1, and they never settle on a single tangent.

Cusp: the one-sided derivatives are both infinite, but of opposite sign. The graph forms a sharp point, like the top of a mountain upside down or upright. Example: $f (x) = ∣ x ∣$ at $x = 0$ :

x \to 0^{+} lim f^{'} (x) = + \infty, x \to 0^{-} lim f^{'} (x) = - \infty

Infinite, opposite signs. Here too, at $x = 0$ there is a minimum.

Inflection point with a vertical tangent: the one-sided derivatives are both infinite but of the same sign. The graph has no point, it has a vertical stretch that the function crosses while continuing to rise (or fall) in the same direction. Example: $f (x) = 3 x$ at $x = 0$ , with

f^{'} (x) = \frac{1}{3 3 x ^{2}}

which tends to $+ \infty$ both from the right and from the left: same sign, so it is neither a maximum nor a minimum, it is an inflection point with a vertical tangent.

The practical point: every time there is a point of non-differentiability, put it in the table together with the zeros of $f^{'}$ . The sign of $f^{'}$ before and after that point tells you anyway whether it is a maximum, a minimum or neither — the formula of the test is identical to that of stationary points, the only change being that there $f^{'}$ is undefined instead of zero.

The sign table of f’: how to build it without mistakes

In the sign table of $f^{'}$ you must insert, in increasing order on the $x$ row:

the ends of the domain;
the points excluded from the domain (vertical asymptotes, holes);
the zeros of $f^{'} (x)$ (the stationary points);
the points of non-differentiability (corners, cusps, inflection points with a vertical tangent).

An example of structure, for a function with domain $R ∖ {1}$ and $f^{'}$ zero at $x = 0$ and $x = 2$ :

$x$	$- \infty$		$0$		$1$		$2$		$+ \infty$
sign of $f^{'} (x)$		$+$	$0$	$-$	undefined	$-$	$0$	$+$
behaviour of $f$		increasing	max	decreasing	asymptote	decreasing	min	increasing

Be careful not to confuse it with the table of the sign of $f$ (the one from part 2, where you study where the graph lies above or below the x axis): they are two different tables, with different purposes, often with the same critical points but a completely different meaning of the sign row. Keeping them separate in your head — and on the page — avoids most of the confusion I see in tests.

Two worked examples

Example 1 — rational function

f (x) = \frac{x ^{2}}{x - 1}, domain: x \neq = 1

Derivative with the quotient rule:

f^{'} (x) = \frac{2 x ( x - 1 ) - x ^{2} \cdot 1}{( x - 1 ) ^{2}} = \frac{2 x ^{2} - 2 x - x ^{2}}{( x - 1 ) ^{2}} = \frac{x ^{2} - 2 x}{( x - 1 ) ^{2}} = \frac{x ( x - 2 )}{( x - 1 ) ^{2}}

The denominator $(x - 1)^{2}$ is always positive (where defined), so the sign of $f^{'}$ coincides with that of the numerator $x (x - 2)$ : positive outside the interval $[0, 2]$ , negative inside.

Table: $f^{'} > 0$ on $(- \infty, 0)$ , $f^{'} < 0$ on $(0, 1)$ , $f^{'} < 0$ on $(1, 2)$ , $f^{'} > 0$ on $(2, + \infty)$ . Note that I write two separate decreasing intervals, $(0, 1)$ and $(1, 2)$ , and not “decreasing on $(0, 2)$ ”: at $x = 1$ the function does not exist (there is the vertical asymptote), so I cannot join the two pieces as if they were connected.

Conclusion: relative maximum at $x = 0$ , with $f (0) = 0$ ; relative minimum at $x = 2$ , with $f (2) = \frac{4}{1} = 4$ . Neither is absolute on the whole domain: for $x \to 1^{+}$ the function tends to $+ \infty$ and for $x \to 1^{-}$ it tends to $- \infty$ (you see it from the limits calculated in part 3), so there is no absolute maximum or minimum at all. The relative maximum at 0 is only the highest point of the left branch, and the relative minimum at 2 only the lowest point of the right branch — and this is where you see well why “relative” and “absolute” are not the same thing.

Example 2 — function with an exponential

f (x) = x e^{- x}, domain: R

Derivative with the product rule:

f^{'} (x) = 1 \cdot e^{- x} + x \cdot (- e^{- x}) = e^{- x} (1 - x)

$e^{- x}$ is always positive, so the sign of $f^{'}$ depends only on $(1 - x)$ : positive for $x < 1$ , negative for $x > 1$ . There are no points excluded from the domain, so here monotonicity can be stated without splitting anything: increasing on $(- \infty, 1)$ , decreasing on $(1, + \infty)$ .

At $x = 1$ the sign goes from $+$ to $-$ : relative maximum, and it is worth $f (1) = 1 \cdot e^{- 1} = \frac{1}{e}$ . Since for $x \to - \infty$ the function tends to $- \infty$ and for $x \to + \infty$ it tends to $0$ (never exceeding $1/ e$ again, given that it is the only stationary point and the function is always decreasing afterwards), this relative maximum is also absolute.

The mistakes I see most often

Studying the value instead of the sign. Substituting numbers into $f^{'} (x)$ to “see what number comes out” instead of setting up the inequality $f^{'} (x) > 0$ . You waste time and get nothing useful for monotonicity.
Forgetting the points of non-differentiability. Concentrating only on the zeros of $f^{'}$ and skipping corners, cusps, inflection points with a vertical tangent — which may very well be a maximum or a minimum even without $f^{'}$ vanishing there.
Declaring a maximum or minimum without checking the sign change. Finding $f^{'} (x_{0}) = 0$ and writing “maximum” or “minimum” straight away without checking what the sign does before and after — and ending up labelling as an extremum a point that is really an inflection point with a horizontal tangent.
Saying “increasing on the whole domain” with a split domain. If there is a vertical asymptote in the middle, the intervals of monotonicity must be listed separately even when the sign of $f^{'}$ is the same on both sides.
Confusing the sign of $f$ with that of $f^{'}$ . They are two different tables with different purposes: one tells you where the graph lies above or below the x axis, the other where it rises or falls. Mixing them in a single row of calculations is the most common source of slips.

How we work on this in lessons

First we train the reflex “I see $f^{'} (x) = 0$ or $f^{'}$ undefined → I check the sign before and after”, on simple functions, until it becomes automatic before moving on to longer ones.
We build every exercise with the complete table — points excluded from the domain, zeros of $f^{'}$ , points of non-differentiability — not just with the final conclusion, because that is where marks are lost in the test.
We work on functions with a split domain on purpose, because that is where almost every badly written “increasing on the whole domain” is born.
Generally 2-3 lessons are enough to fix this step of function study well, once the limits and asymptotes of part 3 are already solid.

The next natural step is concavity: go to part 5: concavity and inflection points. If you want to revisit function study with me, from the basics or only on the weak points, in my maths lessons we always start from where you get stuck: write to me for a free 30-minute introductory call via the contact page.

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 6: the graph and worked examples. To train on the topic in a real test, there is the Year 5 test on derivatives; the part on calculating derivatives 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.

Function study, part 4: monotonicity, maxima and minima