Real Function with Strictly Positive Second Derivative is Strictly Convex

Theorem

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

$\map {f} x > 0$ for each $x \in \openint a b$.

Then $f$ is strictly convex on $\openint a b$.

Proof

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

Since $f > 0$, we have that $f'$ is strictly increasing from Real Function with Strictly Positive Derivative is Strictly Increasing.

$\blacksquare$

Also see

• Second Derivative of Convex Real Function is Non-Negative

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