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_0inA when there is δ>0 for which
f(x)<=f(x_0) , xinAnn(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 xinA 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.
