Given a point #M(x_0,f(x_0))#, if #f# is decreasing in #[a,x_0]# and increasing in #[x_0,b]# then we say #f# has a local minimum at #x_0#, #f(x_0)=...#
If #f# is increasing in #[a,x_0]# and decreasing in #[x_0,b]# then we say #f# has a local maximum at #x_0#, #f(x_0)=....#
More specifically, given #f# with domain #A# we say that #f# has a local maximum at #x_0##in##A# when there is #δ>0# for which
#f(x)<=f(x_0)# , #x##inAnn##(x_0-δ,x_0+δ)# ,
In similar way, local min when #f(x)>=f(x_0)#
If #f(x)<=f(x_0)# or #f(x)>=f(x_0)# is true for ALL #x##in##A# then #f# has an extrema (absolute)
If #f# has no other local extremas in its domain #D_f# then we say #f# has an extrema (absolute) at #x_0#.
Creating a monotony table in each case where you can study #f'# sign and #f# monotony in their domain will make things easier.
