Concavity

Maths

Definition

A function is concave-(down) on if the chord between lies below . This leads to the formal definition

Derivation

1. First we construct the line between getting .
2. Then we can use to get a formula for any point . This is linearly interpolating between and .
3. We want the line to lie below so we get the inequality which is equivalent to .
4. With some basic algebra to clean up on RHS we get as desired

A function is convex-(down) (equivalently concave-up) if the reverse inequality holds e.g. the chord lies above . The exact same derivation as before.