# 7. extrema and increasing and decreasing intervals

A function is said to be increasing in an interval if, for all x1 and x2 in the interval such that x1 < x2, then f(x1) < f(x2).

A function is said to be decreasing in an interval if, for all x1 and x2 in the interval such that x1 < x2, then f(x1) > f(x2).

The maximum and minimum of a function, known collectively as extrema, are the largest and smallest value that the function takes at a point either within a given neighbourhood (local or relative extremum) or on the function domain in its entirety (global or absolute extremum).

If f is derivable in (a,b):
– f is increasing in (a,b) ↔ f’(x) > 0

– f is decreasing in (a,b) ↔ f’(x) < 0

If f is derivable in cЄR, then:

- f has a relative extremum in c → f’(c) = 0

NOTE: the candidates to be relative extrema are cЄR/ f’(c) = 0 or f is not derivable in c

Exercise: find the increasing and decreasing intervals and the extrema of the function y = x·ex

Solutions: minimum (-1,-1/e), increasing (-1,∞): decreasing (-∞,-1)