The Face Isn't the Definition
Every calculus student learns to eyeball a graph and call it "smiley" or "frowny," concave up or concave down. It's a fine mnemonic, but it's not mathematics — it's a description of a picture, not a property you could hope to prove something from. Spivak's Calculus spends a full appendix after Chapter 11 making this precise, and the payoff is worth the trouble: once convexity has an honest definition, you can prove real theorems about where tangent lines sit, why f′ behaves the way it does, and how to locate the exact points where a curve switches its bending direction — not by squinting, but by computation.
Definition 1: A Chord That Never Dips Below the Graph
Here is the actual definition. A function f is convex on an interval if, for all a and b in that interval, the straight line segment joining (a, f(a)) and (b, f(b)) lies above the graph of f. That's it — no derivatives required. Draw any two points on the curve, connect them with a chord, and if that chord never dips below the curve between them, the function is convex there. (Flip "above" to "below" and you get concave; concave functions are exactly the functions of the form −f where f is convex, so every theorem about convexity has a free twin for concavity.)
This geometric statement translates into an algebraic one. The chord between the two points is the graph of
g(x) = [f(b) − f(a)] / (b − a) · (x − a) + f(a),
and after simplifying the inequality g(x) > f(x), Spivak arrives at a cleaner, more useful restatement:
Definition 2. f is convex on an interval if, whenever a < x < b in the interval,
[f(x) − f(a)] / (x − a) < [f(b) − f(a)] / (b − a).
In words: the slope from a up to any intermediate point x is always less than the slope all the way from a to b. Slopes of secant lines are increasing as you slide the right endpoint further out. That single fact is the engine behind everything that follows.
Why It's More Than a Sketching Trick
It would be easy to file convexity under "extra detail for pretty graphs," and Spivak admits as much — the definitions get introduced only after graph-sketching is otherwise complete. But the reason the appendix exists at all is that convexity turns out to control something much sturdier: the relationship between a function and its own tangent lines. Spivak proves, as Theorem 1, that if f is convex and differentiable at a, then the entire graph of f lies above the tangent line at (a, f(a)), touching it only at that one point. And if a < b, then f′(a) < f′(b) — the derivative itself is an increasing function.
That second fact is the hinge that connects convexity to the second derivative. Theorem 2 shows the converse: if f is differentiable and f′ is increasing, then f is convex. Since f″ > 0 is exactly what tells you f′ is increasing, this gives the version of the definition most people actually remember: a twice-differentiable function is convex on an interval wherever its second derivative is positive there. The chord-above-the-graph condition and the tangent-line-below-the-graph condition and the "second derivative positive" condition are three faces of one underlying fact, and Spivak proves the equivalences rather than asserting them — the proofs use nothing fancier than the Mean Value Theorem and a clever auxiliary function, but they are genuinely proofs, not appeals to a picture.
A Worked Example: Where Does f(x) = 1/(1 + x²) Bend?
Spivak works through f(x) = 1 / (1 + x²) as a test case, and it's worth walking through because it shows the machinery earning its keep. First derivative:
f′(x) = −2x / (1 + x²)²,
which is zero only at x = 0, positive for negative x, negative for positive x — so the graph rises to a single peak at (0, 1) and falls away toward zero on both sides, a bell-like shape. So far this is ordinary first-derivative analysis. The second derivative is where convexity enters:
f″(x) = 2(3x² − 1) / (1 + x²)³.
This vanishes exactly at x = ±√(1/3), and since it's continuous, it must keep constant sign on each of the three intervals those two points cut out. Checking one point in each — f″(−1) = 1/2 > 0, f″(0) = −2 < 0, f″(1) = 1/2 > 0 — tells you the whole story: f is convex on (−∞, −√(1/3)) and on (√(1/3), ∞), and concave on the middle stretch (−√(1/3), √(1/3)). The bell shape isn't uniformly curved the way a parabola is; it has genuine convex "wings" flanking a concave "cap," and the two crossover points, where the tangent line actually pierces through the curve rather than staying on one side, are called inflection points.
A Warning Built Into the Definition
It's tempting to think f″(a) = 0 automatically marks an inflection point, but Spivak is careful to block that shortcut with a second example: for f(x) = x⁴, f″(0) = 0, yet f is convex everywhere, and its tangent line at the origin never crosses the graph at all. To actually certify an inflection point, you need f″ to change sign on either side of a — not merely to touch zero there. It's a small distinction, but it's exactly the kind of precision that separates "the graph looks like it bends here" from a claim you can actually defend.
This is the appendix in miniature: a geometric idea (a chord staying above a curve) turned into an algebraic condition, then connected — via genuine proofs, not hand-waving — to the second derivative you already knew how to compute. If you want to see the full chain of theorems, including the trickier converse results and the connection to Jensen's inequality that the problems build toward, it's all laid out in Spivak's Calculus, 4th Edition.
Further reading: How to self-study Spivak's Calculus · Table of contents · Index · Calculus, 4th Edition · Answer Book
Next time you sketch a curve, can you point to where it's convex and prove it — not just see it?
