value

[shashank.ism]

The minimum possible value of the largest of ab, 1-a-b+ab, and a+b-2ab if 0 <= a <= b <=1 is

a 4/9
b 1/9
c 5/9
d 1/3
e 7/9

[ajith]

The minimum possible value of the largest of ab, 1-a-b+ab, and a+b-2ab if 0 <= a <= b <=1 is

a 4/9
b 1/9
c 5/9
d 1/3
e 7/9

1-a-b+ab + a+b-2ab +ab = 1

So the minimum will be when each of these equals 1/3

(1-a)(1-b) =1/3
1-a-b +1/3 =1/3
1-a-b = 0
a+b=1 ab =1/3
x^2 -x +1/3 =0 doesnt have any roots

when a=b =1/3 a minimum of 4/9 (2/3*2/3) is created

[komal]

The minimum possible value of the largest of ab, 1-a-b+ab, and a+b-2ab if 0 <= a <= b <=1 is

a 4/9
b 1/9
c 5/9
d 1/3
e 7/9

This is another possible approach i learnt from complore.com

Let s = a + b, p = ab, so a and b are (s+/- root(s2-4p))/2 . Since a and b are real , s2 - 4p>=0 . If one of the three quantities is less than or equal to 1/9, then at least one of the others is at least 4/9 by the pigeonhole principle since they add up to 1. Assume that s-2p < 4=9, then s2 - 4p < (4/9 + 2p)2 - 4p , and since the left side is non-negative we get 0<= p2 -(5/9) p+4/81 =(p-1/9)(p-4/9).
This implies that either p<=1/9 or p>=4/9 , and either way we're done. This minimum is achieved
if a and b are both 1/3, so the answer is 4/9

[ajith]

The minimum possible value of the largest of ab, 1-a-b+ab, and a+b-2ab if 0 <= a <= b <=1 is

a 4/9
b 1/9
c 5/9
d 1/3
e 7/9

This is another possible approach i learnt from complore.com

Let s = a + b, p = ab, so a and b are (s+/- root(s2-4p))/2 . Since a and b are real , s2 - 4p>=0 . If one of the three quantities is less than or equal to 1/9, then at least one of the others is at least 4/9 by the pigeonhole principle since they add up to 1. Assume that s-2p < 4=9, then s2 - 4p < (4/9 + 2p)2 - 4p , and since the left side is non-negative we get 0<= p2 -(5/9) p+4/81 =(p-1/9)(p-4/9).
This implies that either p<=1/9 or p>=4/9 , and either way we're done. This minimum is achieved
if a and b are both 1/3, so the answer is 4/9

Now we have source, OA and OE and do not tell me this is GMAT question

[komal]

Now we have source, OA and OE and do not tell me this is GMAT question

Yes this does not seem like a gmat question but it was nice to learn something new.

[ajith]

Now we have source, OA and OE and do not tell me this is GMAT question

Yes this does not seem like a gmat question but it was nice to learn something new.

Since you have brought it up, Could you explain me the solution, that you have learned? I do not want to discourage anybody but, copy pasting something from somewhere may not help anybody.