Concavity and inflection points

A curve is concave up (∪) on an interval where its slope $f'$ is strictly increasing, and concave down (∩) where it is strictly decreasing. For differentiable f, concave up also means that the tangent turns counterclockwise as x increases, and that the curve lies above every tangent and below every chord in the interval, apart from the points where they meet. Concave down reverses all three. The chord form is the definition many textbooks start from, for f continuous on the interval and any two points $x_1 \ne x_2$ in it:

The upper graph shows f, the tangent at P and short tangents spaced along the curve. The pale yellow gap between the curve and P’s tangent shows how far the curve stays from it. The lower graph draws f″, and the strip between the graphs marks each interval with ∪ or ∩.

If $f'' > 0$ on an interval, then $f'$ is strictly increasing and the curve is concave up there. $f'' < 0$ makes it concave down. The condition is sufficient, not necessary: $x^4$ is concave up on the whole line although $f''(0) = 0$, so the strip describes an interval, not the sign of f″ at each point.

Why does the tangent turn? Its angle φ with the x-axis has $\tan\varphi = f'$, so

This has the sign of f″, so with f″ > 0 the tangent turns counterclockwise. Its size is not f″ alone, because where the curve is steep the same f″ turns the tangent slowly. The yellow arc at P sketches which way the tangent is turning and how fast at this moment, not the angle it turns further along. The curve stays above its tangent $T_a$ at a because $f(x) - T_a(x) = \int_a^x \big(f'(t) - f'(a)\big)\,dt$. While the slope increases, the integrand has the sign of t − a, so the integral is positive on both sides of a.

A point where f is continuous and the concavity is opposite on its two sides is an inflection point (◇). Where the curve has a tangent there, it crosses it: on $x^3 - 3x$ the tangent at 0 is y = −3x, and $f(x) - (-3x) = x^3$ changes sign at 0. If f″ exists at an inflection point it is 0, but $f''(c) = 0$ is not a test. The function $x^4$ has f″(0) = 0 and no inflection, while $\sqrt[3]{x}$ has no f′ or f″ at 0, yet turns from ∪ to ∩ there.

A flat tangent with the curve above it on both sides is a valley. If $f'(c) = 0$ and $f''(c) > 0$, f has a local minimum at c, and if $f''(c) < 0$, a local maximum.

The curve is sure to stay on one side of a tangent only within the tangent’s own interval. On $x^3 - 3x$, the flat tangent at x = 1 meets the curve again at x = −2, and left of that point the curve lies below it. The tangent is solid within the concave-up interval x ≥ 0 and dotted outside it.

Books name the shapes differently. In English ∪ is convex or concave up, and ∩ concave or concave down. Chinese textbooks use 凹 and 凸 both ways round. The page labels the shapes ∪ and ∩, which do not change with the name.