DerivativeApplications

# 7. extrema and increasing and decreasing intervals

A function is said to be **increasing** in an interval if, for all x_{1} and x_{2} in the interval such that x_{1} < x_{2}, then f(x_{1}) < f(x_{2}).

A function is said to be **decreasing** in an interval if, for all x_{1} and x_{2} in the interval such that x_{1} < x_{2}, then f(x_{1}) > f(x_{2}).

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 cЄ

**R**, 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·e

^{x}

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

Licensed under the Creative Commons Attribution Non-commercial Share Alike 3.0 License