“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.

YONG HUANG

YONG HUANG

Hi, I'm Yong Huang. I've recently graduated from Cornell Tech and obtained my master's degree, I shall start my Ph.D. in Computer Science this fall at UC Irvine. Thank you for visiting my site.