Question

Which of the following equations MUST be true, given that x is an integer ?
I. $x^3\ge {(-x)}^3$
II. $\frac{x^3}{x}\ge \frac{x^2}{x}$
III. $x(x+1)\le -x+x^3$

• A. I only
• B. II only
• C. III only
• D. I and III only
• E. I, II, and III

Answer 1

Answer: (B)

II only

$For\left(I\right)$ $x^3\ge {-x}^3$

For $x>0,x^3>{(-x)}^3$

For $x<0,x^3<{(-x)}^3$

For $x=0,x^3={(-x)}^3$

$\Rightarrow x^3\ge {-x}^3$ for $x>0$

$For\left(II\right)$ $\frac{x^3}{x}\ge \frac{x^2}{x}$

$1>x>0,x>x^2$

$\frac{x^2}{x}\ge \frac{x^3}{x}$

$x\ge 0,x\in I,x^2\ge x$

$\frac{x^3}{x}\ge \frac{x^2}{x}$

$x\le 0,x\in I,x^3\le x^2$

$\frac{x^3}{x}\ge \frac{x^2}{x}$

$For\left(III\right)$$x(x+1)\le -x+x^3$

$x^3-x-x^2-x\ge 0$

$x(x^2-x-2)\ge 0$

$x(x-2)(x+1)\ge 0$

$x\in [-1,0]\cup [2,\infty ]$

Only $II$ is true.