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?

## Comments

Sort by:

TopNewestI 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 · 1 year, 10 months ago

Log in to reply

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, 3 months ago

Log in to reply

– Nanayaranaraknas Vahdam · 2 years, 3 months ago

In your example, 0,1 have a non-negative product.Log in to reply

– Nanayaranaraknas Vahdam · 2 years, 3 months ago

I changed the third one, but the first one is correct.Log in to reply