Question

The function $f\left(x\right)=(x/3)+(3/x),x\ne 0$decreases in the interval.

• A. $(-\infty ,3)$
• B. $(-3,3)$
• C. $(3,\infty )$
• D. $(-9,9)$

Answer 1

The correct option is B

$(-3,3)$

Evaluating for the correct answer

$f\left(x\right)=(x/3)+(3/x),x\ne 0$

$$\begin{gathered} \Rightarrow f\text{'}\left(x\right)=\frac{d\left({\displaystyle \frac{x}{3}}+{\displaystyle \frac{3}{x}}\right)}{dx} \\ \Rightarrow f\text{'}\left(x\right)=\frac{1}{3}-\frac{3}{x^2} \end{gathered}$$

For, decreasing function $f\text{'}\left(x\right)<0$

$$\begin{gathered} \Rightarrow \frac{1}{3}-\frac{3}{x^2}<0 \\ \Rightarrow \frac{1}{3}<\frac{3}{x^2} \\ \Rightarrow \frac{1}{9}<\frac{1}{x^2} \\ \Rightarrow x^2<9 \\ \Rightarrow -3<x<3 \end{gathered}$$

Hence, option B is correct.