The function $f(x) =\frac{x}{2}+\frac{2}{x},x≠0$ is increasing on

(A) (-∞, -2)
(B) (-2, 2)
(C) (2, ∞)
(D) (-1, 1)

Choose the correct answer from the options given below:

(A) and (C) only

The correct answer is Option (3) → (A) and (C) only

$f(x)=\frac{x}{2}+\frac{2}{x},\;x\ne0$

$\frac{df}{dx}=\frac{1}{2}-\frac{2}{x^2}$

Increasing when $\frac{df}{dx}>0$

$\frac{1}{2}-\frac{2}{x^2}>0$

$\frac{1}{2}>\frac{2}{x^2}$

$x^2>4$

$|x|>2$

Hence increasing for $x<-2$ and $x>2$

From the given options this corresponds to

(A) $(-\infty,-2)$ and (C) $(2,\infty)$

The function is increasing on $(-\infty,-2)$ and $(2,\infty)$.