# Second Derivative of Convex Real Function is Non-Negative

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## Theorem

Let $f$ be a real function which is twice differentiable on the open interval $\openint a b$.

Then $f$ is convex on $\openint a b$ if and only if its second derivative $f'' \ge 0$ on $\openint a b$.

## Proof

From Real Function is Convex iff Derivative is Increasing, $f$ is convex if and only if $f'$ is increasing.

From Derivative of Monotone Function, $f'$ is increasing if and only if its second derivative $f'' \ge 0$.

$\blacksquare$

## Also see

- Second Derivative of Concave Real Function is Non-Positive
- Twice Differentiable Real Function with Negative Second Derivative is Strictly Concave

## Sources

- 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 12.19$