Convexity and second derivative: f'' > 0 ⇒ f convex

The key link

Let $f$ be twice differentiable on $I$.

• $f''(x) \geq 0$ on $I$ ⇒ $f$ convex on $I$ (curve above its tangents).
• $f''(x) \leq 0$ on $I$ ⇒ $f$ concave on $I$ (curve below its tangents).

Memory hint: convex = “cup shape” (like $x^2$), concave = “cap shape” (like $-x^2$).

Inflection point

An inflection point is where the curve changes convexity.

It occurs where $f''$ changes sign (often at $x_0$ with $f''(x_0) = 0$).

⚠️ $f''(x_0) = 0$ is not enough: you need a SIGN CHANGE.

Classic examples

Function	Convexity
$x^2$	convex everywhere ($f'' = 2 > 0$)
$-x^2$	concave everywhere
$e^x$	convex everywhere ($f'' = e^x > 0$)
$\ln x$	concave on $(0, +\infty)$
$x^3$	concave on $(-\infty, 0]$, convex on $[0, +\infty)$, inflection at $0$

Alternative: $f'$ monotone

• $f$ convex $\iff$ $f'$ increasing.
• $f$ concave $\iff$ $f'$ decreasing.

Application: convexity inequality

If $f$ is convex on $I$, for all $a, b \in I$ and $t \in [0, 1]$:

$f\big(t a + (1-t) b\big) \leq t f(a) + (1-t) f(b)$

The chord is above the curve.

Method for an inflection point

1. Compute $f''(x)$.
2. Solve $f''(x) = 0$.
3. Check that $f''$ changes sign at this point.

Common mistakes

1. Confusing convex and increasing (no direct link).
2. Writing ”$f$ changes convexity at $x_0$” without checking the sign change of $f''$.
3. Forgetting that $f$ must be twice differentiable to speak of $f''$.

?À toi de jouer

f(x) = e^x. Nature of f?

A.Convex on ℝB.Concave on ℝC.Convex on [0, ∞) onlyD.Inflection point at 0

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