Question

Let $f:[-3,3]\to R$ where $f\left(x\right)=x^3+\sin x+\left[\frac{x^2+2}{a}\right]$ be an odd function then value of $a$ is (where $[.]$ represent greatest integer function and $a$ is positive)

• A. less than $11$
• B. $11$
• C. greater than $11$
• D. none of these

Answer 1

Answer: (C)

greater than $11$

Given $f\left(x\right)=x^3+\sin x+\left[\frac{x^2+2}{a}\right]..\left(i\right)$
$\therefore f(-x)=-x^3-\sin x+\left[\frac{x^2+2}{a}\right]..\left(ii\right)$
Also, given $f(-x)=-f\left(x\right)$
$\therefore f\left(x\right)+f(-x)=0$
$\Rightarrow 2\left[\frac{x^2+2}{a}\right]=0$ (by (i) and (ii))
$\Rightarrow 0\le \frac{x^2+2}{a}<1\forall -3\le x\le 3$

As $x^2+2>0$, we need $a>0$.

Now, $x^2+2<a\Rightarrow a>x^2+2\forall x\in [-3,3]$
$\therefore a>11$ ($\because$ Maximum value of $x^2+2$ in $-3\le x\le 3$ is $11$)