To return month where price is lowest for product from A2
, use this array formula (if there're more then one month with lowest price - formula will return first one):
=INDEX(F:F,MATCH(MIN(IF(B:B=A2,E:E)),IF(B:B=A2,E:E),0))
note, since it's an array formula, you should press CTRL+SHIFT+ENTER to evaluate it