Question

Consider $f\left(x\right)=\{x+[{\log }_2(2+x)\left]\right\}+\{x+[{\log }_2(2+x^2)\left]\right\}+...+\{x+[{\log }_2(2+x^{10})\left]\right\},$ then correct statement is
(where {} and [] denotes the fractional part function and greatest integer function respectively)

• A. $\left[f\right(e\left)\right]=7$
• B. $f\left(\pi \right)=20\pi -60$
• C. Number of solutions of the equation $f\left(x\right)=x$ is $9$
• D. Number of solutions of the equation $f\left(x\right)=x$ is $10$

Answer 1

Answer: (A), (C)

Number of solutions of the equation $f\left(x\right)=x$ is $9$

$\{x+[{\log }_2(2+x)\left]\right\}=\{x+[{\log }_2(2+x^2)\left]\right\}=....=\{x+[{\log }_2(2+x^{10})\left]\right\}=x$
$\therefore f\left(x\right)=\left\{x\right\}+\left\{x\right\}+----+\left\{x\right\}$
$f\left(x\right)=10\left\{x\right\}$
$f\left(e\right)=10\left\{e\right\}=10(e-2)$
$\therefore \left[f\right(e\left)\right]=7$
$f\left(\pi \right)=10(\pi -3)$
$f\left(x\right)=x$

So total number of solutions = $9$