- #1
economicsnerd
- 269
- 24
[I asked this question over a year ago, but I thought I'd try again.]
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 and makes no reference to differentiation.
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 and makes no reference to differentiation.