We can get a lot of good intuition for how first and second derivatives work by interpreting a sign restriction.(adsbygoogle = window.adsbygoogle || []).push({});

Let ##I\subseteq \mathbb R## be an interval and ##f:I\to\mathbb R##.

1) If ##f## is differentiable, then ##f## is monotone iff ##f'\geq 0## everywhere.

2) If ##f## is twice differentiable, then ##f## is convex iff ##f''\geq 0## everywhere.

Monotonicity and convexity are very nice for a couple reasons:

- They're easy to interpret/visualize. e.g. Convexity means all secants lie above the graph.

- They're both global properties.

- They're both easy to state without having defined a derivative.

Is there a similarly interpretable condition which is equivalent (when ##f## happens to be thrice differentiable) to ##f'''\geq0##? Ideally, I'd be interested in a condition which (like monotonicity and convexity) is global and makes no reference to differentiation.

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# Intuition for positive third derivatives

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