Question

The possible values of $\frac{1}{x^2+3}$ lie in the interval

• A. $[\frac{-1}{6},0)$
• B. $(0,\frac{1}{6})$
• C. $(0,\frac{1}{3}]$
• D. $[\frac{-1}{3},0)$

Answer 1

The correct option is C

$(0,\frac{1}{3}]$

We know that $x^2\ge 0\forall x\in R$

$\Rightarrow x^2+3\ge 3;\forall x\in R$

$3\le x^2+3<\infty$

$\Rightarrow \frac{1}{3}\ge \frac{1}{x^2+3}>\frac{1}{\infty }$

$\Rightarrow 0<\frac{1}{x^2+3}\le \frac{1}{3}$

So, the possible values of $\frac{1}{x^2+3}$ lie in the interval $(0,\frac{1}{3}]$