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The minimum possible value of the largest of $ab$, $1-a-b+ab$, $a+b-2ab$ provided $0\leq a\leq b\leq 1$ is:

$1/3,\quad 4/9,\quad 5/9,\quad 1/9,\quad\text{ or }\quad \text{NONE}$

My approach :

I worked out that $\,1-a-b+ab\,$ will be the largest. Now $\,1-a-b+ab = (1-a)(1-b)\,$.

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(1-a-b+ab)-(ab)=1-a-b which is not necessarily positive for the given conditions(take a=0.6,b=0.7). – Aang Jun 27 '12 at 11:57
Some trolls are downvoting without any explanation. I'd rather delete my answer... – DonAntonio Jun 27 '12 at 12:14
Well each of the expressions is positive (since e.g. $a\geq ab$) so the minimum is bounded below by zero. Try some special cases ($a=b=\frac 1 2 , \text { or } a=b=\frac 1 3\text { or } a = \frac 1 3, b=\frac 2 3$ (taking thirds because of the answers suggested) and see what emerges. – Mark Bennet Jun 27 '12 at 12:18
The sum $1 = ab + (1 - a - b + ab) + (a + b - 2ab)$ is at most $3$ times the maximum, with equality only where all are equal, which can't ever happen for real $a$ and $b$. So the largest of the three will always be greater than $1/3$.. – Cocopuffs Jun 27 '12 at 12:54
Where does this problem come from? The answer is surprisingly hard given that it is a multiple choice question. – Simon Markett Jun 27 '12 at 15:01
up vote 1 down vote accepted

Let $M(a,b)=\max\{ab,1-a-b+ab,a+b-2ab\}$, and observe that $$1=ab+(1-a-b+ab)+(a+b-2ab)\leq 3M(a,b),$$ so $$\frac{1}{3}\leq M(a,b)$$ for all such $a,b$. Hence, $\frac{1}{9}$ is ruled out. Can a value of $\frac{1}{3}$ actually be achieved? Well (as Cocopuffs pointed out), it can only be achieved if $\frac{1}{3}=ab=1-a-b+ab=a+b-2ab$, so let's see if that can happen.

Now, $ab=1-a-b+ab$ iff $1-a-b=0$ iff $a+b=1$. Hence, we'd need $a+b=1$ and $ab=\frac{1}{3},$ but this system yields $3a^2-3a+1=0$, which quadratic has a negative discriminant, and so no real solutions. Hence, there are no such real $a,b$, so $\frac{1}{3}$ will never be achieved, and we can rule it out as an answer.

If you try the special cases suggested by Mark Bennet, you'll see that it is possible for $M(a,b)=\frac{4}{9}$, so $\frac{5}{9}$ is ruled out as an answer. The only remaining question is: can we do better than $\frac{4}{9}$? Turns out that's the best we can do.

I really can't see a nice precalculus way to show that the answer is, in fact, $\frac{4}{9}$. Through methods of calculus, if we break $M(a,b)$ down piecewise, we can determine that it is minimized along the curves where the surface has a "fold", and then find that minimum explicitly. (See these W|A graphs for the idea. You'll want to click "Show contour lines" on the 3D plot.)

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