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Some Things to Ponder

Could someone prove these for me:

  1. Given any three real numbers, there exists a pair of two of these numbers such that their product is non-negative.

  2. Given 5 points on a plane, it is impossible to join every point to every other point without two lines intersecting.

  3. Given a function of the form \((x+a)(x+b)\), the minimum is obtained when \(x=\frac{-a-b}{2}\), where \(a\neq-b\)

I had proved the last one, but when \(a=-b\), the result becomes minimum. Could someone explain how this happens?

Note by Nanayaranaraknas Vahdam
2 years, 6 months ago

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I believe for 2 you have the condition no 3 points are colinear. and also,you can generalize the statement to "two paths intersecting", instead of lines. Rutwik Dhongde · 2 years ago

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The first one is false. Counter-example: -1,0,1. And number 3 is false. Counter-example: a=b=0 results in the function x^2, which has no maximum. As for number 2, I know nothing. Leonard Kho · 2 years, 6 months ago

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@Leonard Kho In your example, 0,1 have a non-negative product. Nanayaranaraknas Vahdam · 2 years, 6 months ago

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@Leonard Kho I changed the third one, but the first one is correct. Nanayaranaraknas Vahdam · 2 years, 6 months ago

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