jmc

Question

Let x be a real number. Consider the following five statements: 0 < x^2 < 1 x^2 > 1 -1 < x < 0 0 < x < 1 0 < x - x^2 < 1 What is the maximum number of these statements that can be true for any value of x?

Accepted Answer

One the first two statements, at most one of them can be true (

x^{2}

cannot be both less than 1 and greater than 1). Of the next two statements, at most one of them can be true (

x

cannot be both less than 0 and greater than 0). Hence, at most three statements can be true.

For

0 < x < 1,

the first, fourth, and fifth statements are true, so the maximum number of statements that can be true is

3 .