Q58 / 125 · BITSAT 2020 · Mathematics

Consider the function $f(x)=|x-1| /x^{2},$ then $f(x)$ is

  • A
    increasing in (0,1)$\cup(2, \infty)$
  • B
    increasing in $(-\infty, 0) \cup(1,2)$
  • C
    decreasing in (0,1)$\cup(2, \infty)$
  • Correct answer
    D
    decreasing in (0,1)$\cup(2, \infty)$

Solution

$f(x)=\frac{|x-1|}{x^{2}}=\left\{\begin{array}{l}\frac{1-x}{x^{2}}, x<1, x \neq 0 \\ \frac{x-1}{x^{2}}, x>1\end{array}\right.$ Clearly, $f(x)$ is continuous for all $x \in R$ except at $x=0$ $f^{\prime}(x)=\left\{\begin{array}{l} \frac{x-2}{x^{3}}, x<1, x \pm 0 \\ \frac{2-x}{x^{3}}, x>1 \end{array}\right.$ $f^{\prime}(x)>0 \Rightarrow x<0$ or $1<x<2$ $f^{\prime}(x)<0 \Rightarrow 0<x<1$ or $x>2$ Hence, $f(x)$ is increasing in $(-\infty, 0) \cup(1,2)$ and decreasing in (0,1)$\cup(2, \infty)$.

Hence, the correct answer is option: 'D'.

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