Expectation, variance and generating functions

The function \(f : [0, 2 \pi] \to \mathbb{C}\), \(f(t) = \exp(i t)\), satisfies \(f(0) = f(2 \pi) = 1\) and is \(C^\infty\) on \([0, 2 \pi]\) with \(f'(t) = i \exp(i t)\) (Ex7.1). But \(|f'(t)| = 1\) everywhere, so \(f'\) never vanishes --- Rolle fails. As a consequence, no value of \(c \in {]}0, 2 \pi[\) satisfies \(f(2\pi) - f(0) = f'(c) \cdot 2 \pi\) (the left side is \(0\), the right side has modulus \(2 \pi\)), so the equality form of the mean value theorem fails too.

T7.1 is admitted in this chapter. The inequality results from a simple integral upper bound, justified later in the Integration section. The full proof will be given in Integration on a segment (via \(f(y) - f(x) = \int_x^y f'(t) dt\) and the triangle inequality for integrals). Until then, T7.1 is used as a black box.

Convexity captures, in a single inequality, the geometric idea of a curve that always lies below its chords. From this one definition flows a toolbox: a slope-monotonicity characterization, Jensen's inequality (finite-point version of convexity), a derivative-based test (\(f'\) nondecreasing or \(f'' \ge 0\)), and the dual « graph above its tangents » statement that fuels classical bounds like \(e^x \ge 1 + x\), \(\ln(1+x) \le x\), and \(\sin x \ge 2x/\pi\) on \([0, \pi/2]\). The chapter closes with a brief, scoped treatment of inflection points.
Throughout, \(I\) denotes an interval of \(\mathbb{R}\) (possibly unbounded, possibly closed or open). The duality « concave \(\Leftrightarrow\) \(-f\) convex » is stated once at the start and used implicitly thereafter; concave inequalities reverse the convex ones. Jensen's inequality is stated and proved for finite convex combinations; the H\"older and Minkowski inequalities belong to later integration / normed-spaces material. Inflection points are introduced in the closing section as a safe sufficient criterion.

The geometric idea: a function is convex if its graph stays below every chord drawn between two of its points. We translate this into the chord inequality, then show two characterizations: monotonicity of slopes and Jensen's inequality for finite convex combinations.

Side by side: a parabola \(y = \tfrac{1}{2} x^2\) (convex, graph below the chord between any two of its points) and a (translated) logarithm \(y = \ln x + 0.3\) (concave, graph above the chord between any two of its points). The vertical scaling and shift are cosmetic --- convexity/concavity is preserved. The convex graph is below the chord \(AB\); the concave graph is above the chord \(AB\).

The graph of a convex function lies below its secant on \([x, y]\) and above the secant outside \([x, y]\).

Jensen's inequality is the finite-point version of convexity: it extends the two-point chord inequality of D1.1 to any finite convex combination. We stay within finite combinations only --- any general development on barycentres is out of program.

A convex function on an open interval is continuous. This useful theorem is admitted here.
(The proof uses the slope-monotonicity characterization P1.2 plus the monotone-limit theorem of Limits and continuity.)

When \(f\) is (twice) differentiable, convexity has clean derivative-level characterizations: \(f\) convex \(\Leftrightarrow\) \(f'\) nondecreasing \(\Leftrightarrow\) graph above all tangents. The \(C^2\) version --- \(f\) convex \(\Leftrightarrow\) \(f'' \ge 0\) --- gives the operational test used everywhere. The tangent inequality is the engine behind the classical bounds \(e^x \ge 1 + x\), \(\ln(1+x) \le x\), \(\sin x \le x\), and \(\sqrt{x} \le (x+1)/2\).
Derivative convention. In derivative statements below, either \(I\) is open, or the assertions involving \(f'(a)\) are read for interior points \(a \in \mathring{I}\). When an endpoint is used in an example, the function is understood as the restriction of a differentiable function defined on a larger open interval.

For \(f\) concave differentiable, the inequalities reverse. In particular, the graph of \(f\) lies below every one of its tangents: $$ \forall a, x \in I, \qquad f(x) \le f(a) + f'(a)(x - a). $$ This is the engine behind the bounds \(\ln(1+x) \le x\), \(\sqrt{x} \le (x+1)/2\), and \(\sin x \le x\) proved below.

A function \(f : I \to \mathbb{R}\) is strictly convex on \(I\) if, for every \(x \ne y\) in \(I\) and every \(\lambda \in {]}0, 1[\), \(f((1 - \lambda) x + \lambda y) < (1 - \lambda) f(x) + \lambda f(y)\) (strict inequality). For differentiable functions, \(f'\) strictly increasing on \(I\) implies \(f\) strictly convex. For \(f \in C^2\), the condition \(f'' > 0\) on \(I\) is sufficient but not necessary --- the function \(x \mapsto x^4\) is strictly convex on \(\mathbb{R}\) yet \(f''(0) = 0\).

For a \(C^2\) function satisfying the convex/concave-transition definition above, \(f''(a) = 0\) is necessary but not sufficient. The standard counterexample is \(f(x) = x^4\) at \(a = 0\): \(f''(x) = 12 x^2\) vanishes at \(0\) but is nonnegative everywhere, so \(f\) is convex on all of \(\mathbb{R}\) --- there is no convex/concave transition, hence no inflection point. The exo file drills this pitfall explicitly.

The cubic \(f(x) = x^3 - 3 x\) has \(f''(x) = 6 x\), so the unique inflection point is at \(x = 0\). The graph is concave on \(]-\infty, 0]\) and convex on \([0, +\infty[\).

The integers \(\mathbb{Z}\) form, under their usual addition and multiplication, a self-contained world rich enough to host an entire theory of divisibility. This chapter develops that theory in five steps. We start from the divisibility relation \(b \mid a\) and the Euclidean division theorem \(a = bq + r\), the unique algorithmic device that converts the question « does \(b\) divide \(a\)? » into the question « is the remainder zero? ». We then build on Euclidean division to construct the greatest common divisor \(a \wedge b\), both via the iterative algorithm of Euclid and via the linear-combination identity of Bézout, \(a \wedge b = a u + b v\). We extract the special case \(a \wedge b = 1\) (coprime integers) and prove Bézout's theorem and Gauss's lemma --- the two engines that drive every divisibility argument in the rest of the chapter. We introduce the prime numbers, prove their infinitude (Euclid) and the existence/uniqueness of the prime factorization \(n = \prod_p p^{v_p(n)}\), and use the \(p\)-adic valuation \(v_p\) to give clean closed-form expressions for \(a \wedge b\) and \(a \vee b\). We close by returning to the divisibility relation in a different guise --- congruences modulo \(n\) --- and use the previous machinery to characterize invertibility modulo \(n\) (\(a \wedge n = 1\)) and to prove Fermat's little theorem \(a^p \equiv a \pmod p\) for \(p\) prime.
The approach throughout is elementary: by convention in this chapter, we do not call upon the language of algebraic structures. We therefore avoid, deliberately, all references to groups, rings, ideals, and morphisms; the set \(\mathbb{Z}/n\mathbb{Z}\) appears in the section on congruences as a set of \(n\) classes only, with an explicit reminder that its ring structure is hors programme and will be revisited in the chapter Algebraic structures. Two further chapters extend what we do here: Arithmetic of polynomials replays the entire story for \(\mathbb{K}[X]\) in place of \(\mathbb{Z}\) (Euclidean division, gcd, Bézout, Gauss, irreducibility), and Algebraic structures re-frames divisibility, ideals, and modular arithmetic in the abstract.
Standing convention. Throughout this chapter, a divisor is always a non-zero integer. Whenever we write « \(d \mid a\) », we silently assume \(d \in \mathbb{Z}^*\). In particular \(\operatorname{div}(0) = \mathbb{Z}^*\) (every non-zero integer divides \(0\)). We use \(\mathbb{N} = \{0, 1, 2, \dots\}\) and \(\mathbb{N}^* = \mathbb{N} \setminus \{0\}\). The notations \(a \wedge b\) (greatest common divisor) and \(a \vee b\) (least common multiple) are standard.

Two definitions, one theorem. The definition of divisibility is the entry point of the chapter; Euclidean division is the algorithmic tool that turns « \(b \mid a\)? » into « is the remainder zero? ». The Grade 12 « Spécialité » baseline already covers both --- we recall them quickly, then state and prove rigorously and extend the Euclidean division statement to negative dividends.

Euclidean division turns the divisibility question into an algorithmic one. The Grade 12 statement covers \(a \in \mathbb{N}\), \(b \in \mathbb{N}^*\); we extend to \(a \in \mathbb{Z}\), with the same divisor \(b \in \mathbb{N}^*\) and the same remainder constraint \(0 \leq r < b\). The negative case is the classical trap: students must remember that \(r\) is always non-negative.

The greatest common divisor (gcd) of two integers \(a\) and \(b\) is, intuitively, the largest integer that divides both. Two subtleties stand out: (i) « largest » needs to be qualified --- largest in the natural order \(\leq\) on \(\mathbb{N}^*\) vs.\ largest in the divisibility order \(\mid\) ---, and the gcd turns out to be the largest in both senses; (ii) the gcd is computed effectively by Euclid's algorithm, an iterated Euclidean division. We then prove the central identity of the chapter: Bézout's identity \(a \wedge b = a u + b v\) for some \(u, v \in \mathbb{Z}\). From there the « common-divisors-divide-the-gcd » characterization falls as a corollary, and the lcm \(a \vee b\) (and the formula \((a \wedge b)(a \vee b) = |a b|\)) follow naturally.

The algorithmic computation of the gcd rests on a single observation: when one integer is reduced modulo the other, the gcd doesn't change. Iterated, this rule gives the algorithm of Euclid, and the gcd is read off as the last non-zero remainder.

Bézout's identity is the linear-combination side of the gcd: \(a \wedge b\) writes as \(a u + b v\) for some \(u, v \in \mathbb{Z}\). The proof reuses the Euclid sequence: each remainder \(r_k\) writes as a \(\mathbb{Z}\)-linear combination of \(a\) and \(b\), by induction. Practically, one runs Euclid's algorithm forward and then back-substitutes to extract a pair \((u, v)\).

With Bézout's identity in hand, the characterization of the gcd by its set of divisors is now an immediate corollary.