Second Derivative of Convex Real Function is Non-Negative

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

• Real Function with Strictly Positive Second Derivative is Strictly Convex

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

Sources

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