1. Functions - extreme points.

Consider a real valued function

where I is an open interval in R. We say that f has a local minimum in the point x0 in I if

for all x in some neighborhood to x0 in Ilocal maximum is defined in a corresponding way.

It is well known that if f is differentiable, a necessary condition for x0 to be an extreme point (local maximum or minimum) is

Note that the conversion in general does not hold, but if f also fulfills

then f has a local minimum (maximum) in x0 .

Remark: A point x0 that fulfills

is called a stationary point of f.