Try this...

Data Range
A
B
C
D
E
F
1
Mark
6
300
Mark
200
2
Mark
6
200
3
John
6
10
4
Mark
5
3
5
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This array formula** entered in F1:

=MIN(IF(A1:A4=E1,IF(B1:B4=MAX(IF(A1:A4=E1,B1:B4)),C1:C4)))

** array formulas need to be entered using the key
combination of CTRL,SHIFT,ENTER (not just ENTER).
Hold down both the CTRL key and the SHIFT key
then hit ENTER.