# “Concave and Convex function”

## Definition

A real-valued function \(f\) on an interval (or, more generally, a convex set in vector space) is said to be concave if, for any \(x\) and \(y\) in the interval and for any \(\alpha \in [0,1]\).

[f((1-\alpha) x+\alpha y) \geq(1-\alpha) f(x)+\alpha f(y)]

A function is called strictly concave if

[f((1-\alpha) x+\alpha y)>(1-\alpha) f(x)+\alpha f(y)]

for any \(\alpha \in [0,1]\) and \(x \neq y\)

For a function \(f: \mathbb{R} \rightarrow \mathbb{R}\), this second definition merely states that for every \(z\) strictly between \(x\) and \(y\), the point \((z,f(z))\) on the graph of \(f\) is above the straight line joining the points \((x,f(x))\) and \((y,f(y))\).

Convex is the opposite of concave case.