Slopes of a convex function are increasing

Slopes of a convex function are increasing
Tag	`0BY5FTHEEE`
URI	`m2://0BY5FTHEEE`
Kind	statement, proposition
Writing	W0 stub
MSC	26A51
Status	This copy of the site has no build data. Run `m2 status 0BY5FTHEEE`.
Uses	Convex function

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Statement

Let $I$ be an interval and let $f \colon I \to \mathbb{R}$ be convex. Then for all $x < y < z$ in $I$ , $\frac{f(y) - f(x)}{y - x} \le \frac{f(z) - f(x)}{z - x} \le \frac{f(z) - f(y)}{z - y}.$

Remarks

Write $y = t x + (1 - t) z$ with $t = (z - y)/(z - x)$ , and apply the definition of convexity. The converse holds: a function with this property is convex. The first and the last quotient give the adjacent form ConvexOn.slope_mono_adjacent of Mathlib.

Explanations

L1 human

For a convex function, chords further to the right are steeper. So the slopes of the graph increase from left to right.

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Convex function definition
Slopes of a convex function are increasing statement

What depends on this

6 nodes use this node, directly or through other nodes.

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