# 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
4 years, 2 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.

- 3 years, 9 months ago

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.

- 4 years, 2 months ago

In your example, 0,1 have a non-negative product.

- 4 years, 2 months ago

I changed the third one, but the first one is correct.

- 4 years, 2 months ago