Q120 / 180 · KCET 2013 · Mathematics
The function $f(x)=\frac{x}{3}+\frac{3}{x}$ decreases in the interval
- Correct answerA$(-3,3)$
- B$(-\infty, 3)$
- C$(3, \infty)$
- D$(-9,9)$
Solution
Given,
$f(x)=\frac{x}{3}+\frac{3}{x}$
On differentiating w.r.t. $x$, we get
$f^{\prime}(x)=\frac{1}{3}-\frac{3}{x^{2}}
$Since, function is decreasing
$\begin{array}{ll}
\text { i.e., } & f^{\prime}(x) < 0 \\
\Rightarrow & \frac{1}{3}-\frac{3}{x^{2}} < 0 \\
\Rightarrow & \frac{1}{3} < \frac{3}{x^{2}} \\
\Rightarrow & x^{2} < 9 \\
\Rightarrow & x \in(-3,3)
\end{array}$
which is the required interval in which function $f(x)$ decreases.
Hence, the correct answer is option: 'A'.