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:
Answer & explanation
Correct answer: option 3
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 $(2,\infty)$.