The correct options are
B My wife's cooking is considered the best in the country.
D The two boys' friendship has stood the test of time.
Apostrophe is used to indicate possession. Apostrophe is used before 's' with singular nouns and after 's' with plural nouns.
Option A: 'Wife' is a singular noun and 'cooking' is that of the 'wife', therefore 'wife's cooking' is correctly punctuated.
Option B: 'Island' is marked with apostrophe followed by 's', but there is nothing mentioned that belongs to the island, therefore the use of apostrophe is not appropriate here.
Option C: 'House' is marked with apostrophe followed by 's' but there is nothing mentioned that belongs to the house, therefore the use of apostrophe is not appropriate here either.
Option D: 'Boys' is a plural noun and 'friendship' belongs to the two boys. Therefore, apostrophe is used appropriately here, ie. boys' friendship.