[I asked this question over a year ago, but I thought I'd try again.](adsbygoogle = window.adsbygoogle || []).push({});

Let ##I\subseteq \mathbb R## be an interval and ##f:I\to\mathbb R## be a ##C^\infty## function.

I have the following characterizations:

1) ##f'\geq 0## everywhere iff ##f## is increasing.

2) ##f''\geq 0## everywhere iff ##f## is convex.

The underlined properties above are very nice for a couple reasons:

- They're easy to interpret/visualize. e.g. An increasing function is one with all secant lines having slope ##\geq 0##; a convex function is one with all secants lying above its graph.

- They're both global properties.

-They're both easy to state without having defined a derivative.That is, I can define an increasing function or a convex function, even if my audience doesn't understand what a derivative is.

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

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# Intuition for sign of third derivative

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