LINUX: How to AWK to find maximums in every unique line

With GNU awk:

awk 'NR==1{print; next}
     {
       note=$NF                            # save note from last field
       NF--                                # remove last field from $0
       name=$0
       if(a[name]<note){a[name]=note}
     }
     END{
       for(i in a){print i,a[i]}
     }' file

Output:

firstname lastname note 
Fred Free 10
Ben Quick 12
Mark Marked 20

See: 8 Powerful Awk Built-in Variables – FS, OFS, RS, ORS, NR, NF, FILENAME, FNR

With your shown samples please try following. Written and tested with GNU awk should work with any awk.

awk '
FNR==1{
  print
  next
}
match($0,/[^0-9]*/){
  val=substr($0,RSTART,RLENGTH)
  sub(/[[:space:]]+$/,"",val)
}
{
  arr[val]=(arr[val]>$NF?arr[val]:$NF)
}
END{
  for(key in arr){
    print key,arr[key]
  }
}
'  Input_file

Explanation: Adding detailed explanation for above.

awk '                                     ##Starting awk program from here.
FNR==1{                                   ##Checking condition if this is first line.
  print                                   ##printing current line here.
  next                                    ##next will skip all further statements from here.
}
match($0,/[^0-9]*/){                      ##using match function to match till any digit comes in line.
  val=substr($0,RSTART,RLENGTH)           ##Creating val which has sub string of current line, with values of RSTART and RLENGTH.
  sub(/[[:space:]]+$/,"",val)             ##Substituting spaces coming at last in val wit NULL here.
}
{
  arr[val]=(arr[val]>$NF?arr[val]:$NF)    ##Creating array with index of val and keeping only maximum value by comparing with last field os current line.
}
END{                                      ##Starting END block of this program from here.
  for(key in arr){                        ##Traversing into arr elemennts now.
    print key,arr[key]                    ##Printing key and its value here.
  }
}
'  Input_file                             ##Mentioning Input_file name here.

$ awk 'NR==1     {print; next} 
                 {k=$1 FS $2} 
       max[k]<$3 {max[k]=$3} 
       END       {for(k in max) print k, max[k]}' file

firstname lastname note
Fred Free 10
Ben Quick 12
Mark Marked 20

another approach

$ sed 1q file; sed 1d file | sort -k1,2 -k3nr | awk '!a[$1,$2]++'

firstname lastname note
Ben Quick 12
Fred Free 10
Mark Marked 20

Using any awk, sort, and cut in any shell on every Unix box:

$ awk '{print (NR>1), $0}' file | sort -k1,1n -k4,4rn | cut -d' ' -f2- | awk '!seen[$1,$2]++'
firstname lastname note
Mark Marked 20
Ben Quick 12
Fred Free 10