Twice Differentiable Real Function with 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 $f'$ is strictly increasing.

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

Also see

 * Second Derivative of Convex Real Function is Non-Negative


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