We began with a “proof” that \(1 = 2\) and three questions: how to know an argument is correct, how to prove inequalities, and what a function really is. Here is what the chapter’s nine questions have taught us.
1. How do you know an argument is correct?
Section 3.1 gave us a checklist. At every step, ask three things: is the calculation right? does the step follow logically? what has been assumed without being said? The errors there were typical. Comparing the values at two stationary points says nothing about which is a maximum. Stationary values do not bound a function on an infinite domain. The fundamental theorem of calculus needs a continuous integrand. And an argument can arrive at a “correct-looking” answer by accident.
The same vigilance ran through the rest of the chapter. Section 3.7 showed by counterexample that dropping a hypothesis (\(\mathrm{f}(0) = \mathrm{f}(\pi) = 0\)) breaks an inequality. In Section 3.8, the assumption that the function is differentiable was essential: without it, Cauchy’s functional equation has wild solutions that nobody can write down. And in Section 3.6 and Section 3.3, the equality conditions told us whether a bound was best possible, which is the kind of question a careful mathematician always asks.
2. How do you prove an inequality?
We met the classical inequalities from several directions, and saw how they are related.
Squares are non-negative. The two-number AM–GM is just \((a - b)^2 \geq 0\). Cauchy’s forward–backward induction built the general case from it (Section 3.2).
Convexity. A concave curve lies above its chords. That one fact gives Jensen’s inequality, and with the logarithm it gives AM–GM in a line; with \(\sin x\) it gives a sharp inequality for triangles (Section 3.3).
Areas. The area under a curve plus the area beside it covers a rectangle. That gives Young’s inequality, which leads to Hölder and Cauchy–Schwarz (Section 3.4).
Shape of a graph. Rewriting \(a^b\) versus \(b^a\) as a comparison of \(\frac{\ln x}{x}\) at two points settled \(\pi^e\) versus \(e^\pi\) (Section 3.5).
Interpolation and the triangle inequality. Writing the derivative in terms of values, then bounding term by term, gave Markov’s inequality for quadratics (Section 3.6).
Test functions. Applying a general inequality to a cleverly chosen function gave \(\pi^2 \leq 10\) and \(\pi \leq \frac{22}{7}\) (Section 3.7).
Choosing among these is a skill that comes with practice. A rough guide: products and sums suggest AM–GM; averages of function values suggest Jensen; a function and its inverse suggest Young; a comparison of two expressions of the same form suggests finding the single function behind them.
3. What is a function, really?
The last two sections answered this in two ways. In Section 3.8, \(e^{kx}\) and \(\tanh kx\) were not given by formulas at all; they were the only differentiable functions obeying certain addition laws. That is a deeper kind of understanding than a formula: it explains why these functions keep appearing, in compound interest and in Einstein’s velocity addition. In Section 3.9 we went further and invented a function, the inverse of \(xe^x\), because the functions we had were not enough to solve a natural family of equations. Every function on your calculator was once new in the same way.
4. A toolkit for rigour and inequalities
| Method | Where you used it |
|---|---|
| Checking every step and every hypothesis | Plausible wrong arguments (3.1) |
| Local versus global behaviour | Stationary points and limits at infinity (3.1) |
| Forward–backward induction | AM–GM (3.2) |
| Padding with extra terms | AM–GM (3.2, 3.3) |
| Jensen’s inequality | Triangles; AM–GM (3.3) |
| Comparing areas | Young’s inequality (3.4) |
| Separating variables into one function | \(\pi^e\) versus \(e^\pi\) (3.5) |
| Triangle inequality, equality cases | Markov’s inequality (3.6) |
| Test functions | \(\pi^2 \leq 10\) (3.7) |
| Functional equations | \(e^x\) and \(\tanh x\) (3.8) |
| Inverse functions and substitutions | The W function (3.9) |
Where next
The calculus of variations, glimpsed in Section 3.7, is the subject of Section 4.7.
The geometry behind \(\tanh\) and its addition law appears in Section 7.6.
Chapter 4 takes the careful attitude of this chapter to integration itself, starting from what an integral actually is.
J. Michael Steele, The Cauchy–Schwarz Master Class: the best introduction to inequalities, built entirely from problems.
Daniel Velleman, How to Prove It: a systematic guide to writing correct proofs.
Michael Spivak, Calculus: rigorous calculus from the ground up; many Cambridge students read it the summer before they arrive.
E. A. Maxwell, Fallacies in Mathematics: for practice in spotting what has gone wrong.
Von Neumann joked that in mathematics you don’t understand things, you just get used to them. There is some truth in it: the precise definitions and careful arguments of this chapter can feel strange at first. But getting used to them is exactly what turns a student who can calculate into a mathematician who can be sure. The proof that \(1 = 2\) will never fool you again; the aim of a mathematics degree is to make every subtler error just as easy to see.