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Consider four positive numbers $x_1 ,x_2,y_1$ and $y_2$ such that $y_1y_2 > x_1x_2$ . Consider the number $ S = (x _1y_2 + x_2y_1 ) − 2x_1x_2$ . The number S is (A) always a negative integer; (B) can be a negative fraction; (C) always a positive number; (D) none of these.

Any hint on how to go about this will be appreciated.

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    Are the numbers naturals ? – 2017-01-12
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    Sorry, nothing about that is mentioned in the question. All number's are positive, but we can't say if they can be 0 as well. – 2017-01-12
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    I am querying about restriction to integers ! – 2017-01-12

2 Answers 2

4

we have by $AM_GM$ $$\frac{x_1y_2+x_2y_1}{2}\geq \sqrt{y_1y_2x_1x_2}$$ since we have $$y_1y_2>x_1x_2$$ we get $$x_1x_2y_1y_2>(x_1x_2)^2$$ thus we get $$\frac{x_1y_2+x_2y_1}{2}\geq \sqrt{x_1x_2y_1y_2}>\sqrt{(x_1x_2)^2}=x_1x_2$$ thus we have $$x_1y_2+x_2y_1-2x_1x_2>0$$

2

We have: $S \ge 2\sqrt{x_1x_2y_1y_2} - 2x_1x_2 > 0$ by AM-GM inequality.