Some Things to Ponder

Could someone prove these for me:

Given any three real numbers, there exists a pair of two of these numbers such that their product is non-negative.
Given 5 points on a plane, it is impossible to join every point to every other point without two lines intersecting.
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?

#Algebra #CoordinateGeometry #Points #Maximaminima

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

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 - 6 years, 5 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.

Leonard Kho - 6 years, 11 months ago

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

Nanayaranaraknas Vahdam - 6 years, 11 months ago

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

Nanayaranaraknas Vahdam - 6 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$