4.1 the extreme value theorem

absolute maximum

let f(x) be defined on a set D containing c. If f(c ) is greater than or equal to all values of x in D then f (c ) is (biggest y- value)

absolute minimum

If f( c) is less than or equal to all values of x in D then f(c ) is the

the extreme value theorem

a continues function of f(x) on a closed and bounded interval [a, b] has an absolute max and absolute min

• absolute extrema can occur on end pts and interior pts

local maximum

suppose c is an interior pt of some interval I for which f(x)is defined. If f(c ) is greater than or equal to f(x) for all x in I then f(c ) is

local minimum

if f(c ) is less than or equal to f(x) for all x in I then f(c ) is

critical point

an interior pt of f(x) for which f”© = 0 or f’ (c ) DNE is called

• note must be interior pt

critical point theorem

if f(c ) is a local extrema then c is a critical point

• if c is cp, f(c ) not necessarily a local extremum

to find absolute extrema

• find cp

• evaluate og equation at cp and end pts

• to determine which values biggest and smallest