Question

Given that a function $f(x)=[x{]}^2+x^2$ , where $[x]$ is the greatest integer less than or equal to $x$ . If $f(x)>25$ , then the value of $x$ is

• A. any real number
• B. a member of the set $\left\{x|x\ge 0\right\}$
• C. a member of the set $\{x|x\le -4$ or $x\ge 4\}$
• D. a member of the set $\{x|x\ge 25$ or $x\le 0\}$

Answer 1

Answer: (C)

We have, $f(x)=[x{]}^2+x^2$
Given, $f(x)>25$
$\therefore [x{]}^2+x^2>25$
$\Rightarrow (x-\{x\}{)}^2+x^2>25$
$[\because x=[x]+\{x\}$ where $0\le \{x\}<1]$
$\Rightarrow x^2+\{x{\}}^2+x^2-2x\{x\}>25$
$\Rightarrow 2x^2-2x\{x\}+\{x{\}}^2>25$
$\Rightarrow 2x(x-\{x\})+\{x{\}}^2>25$
$\Rightarrow 2x[x]+\{x{\}}^2>25$
$\Rightarrow 2x[x]\ge 24[\because 0\le \{x{\}}^2<1]$
$\Rightarrow x[x]\ge 12$, which is possible for $x\ge 4$ or $x\le -4$
$\therefore$ Value of $x$ is the member of the set
$\{x|x\le -4$ or $x\ge 4\}$