MHB Is the Product of Real Numbers Always Larger When Exponentiated?

[NoName3]

For any $a \in \mathbb{R}$, let $a^3$ denote $a \cdot a \cdot a$. Let $x, y \in \mathbb{R}$.

1. Prove that if $x < y$ then $x^3 < y^3$.

2. Prove that there are $c, d \in \mathbb{R}$ such that $c^3 < x < d^3$.

 

[Opalg]

For any $a \in \mathbb{R}$, let $a^3$ denote $a \cdot a \cdot a$. Let $x, y \in \mathbb{R}$.

1. Prove that if $x < y$ then $x^3 < y^3$.

2. Prove that there are $c, d \in \mathbb{R}$ such that $c^3 < x < d^3$.

Hint for 1.: $y^3-x^3 = (y-x)(y^2 + xy + x^2)$. Now show that each of those two factors is positive.

 

[NoName3]

Hint for 1.: $y^3-x^3 = (y-x)(y^2 + xy + x^2)$. Now show that each of those two factors is positive.

Thank you.

Since $x< y$ we have $0<y-x$. So the first factor is positive.

$y^2 + xy + x^2 = (y+\frac{1}{2} x)^2+x^2- \frac{1}{4}x^2 = (y+\frac{1}{2} x)^2+\frac{3}{4}x^2 $

which is positive as well. Therefore if $x< y$ then $x^3 < y^3$ as required.