---
tag: 1GZH50HV78
kind: proof
name: Proof that a function with nonnegative second derivative is convex
msc: [26A51]
proves: 0Z41C17TGP
uses: [026BJBQNGJ, 005JT76HCK, 0Y9D2FJRQ4, 07ZMX4ANJS, 0VV0HSW1ZS]
sources:
  - url: https://proofwiki.org/wiki/Real_Function_is_Convex_iff_Derivative_is_Increasing
---

## Proof

We show that $f'$ is increasing and then use the derivative test for convexity.

1. Since $f$ is twice differentiable on $I$, the function $f' \colon I \to \R$ is differentiable on $I$. Its derivative is the [[second derivative|0Y9D2FJRQ4]] $f''$.
2. A differentiable function is [[continuous|07ZMX4ANJS]], so $f'$ is continuous on $I$.
3. Every point of the open interval $I$ is an interior point of $I$, and $(f')'(x) = f''(x) \ge 0$ there.
4. By the [[nonnegative derivative test|026BJBQNGJ]] applied to $f'$, the function $f'$ is increasing on $I$.
5. The function $f$ is differentiable on $I$ and $f'$ is increasing. By the [[derivative test for convexity|005JT76HCK]], $f$ is [[convex|0VV0HSW1ZS]] on $I$.

## Idea

A nonnegative second derivative means the first derivative never decreases. So the slope of $f$ only grows from left to right, and a function whose slope grows is convex.