Q234 / 270 · NDA 2016 (Phase 2) · Mathematics

Paragraph: Let $f(x)=\left\{\begin{array}{ll}\frac{e^x-1}{x}, & x \gt 0 \\ 0, & x=0\end{array}\right.$ be a real valued function.
Question: Which one of the following statements is correct?

  • A
    $f(x)$ is a strictly decreasing function in $(0, x)$,
  • Correct answer
    B
    $f(x)$ is a strictly increasing function in $(0, x)$,
  • C
    $f(x)$ is neither increasing nor decreasing in $(0, x)$
  • D
    $f(x)$ is not decreasing in $(0, x)$.

Solution

$\begin{aligned} & f(x)=\frac{e^x-1}{x} \gt 0 \\ & \Rightarrow f^{\prime}(x)=\frac{x e^x-\left(e^x-1\right)}{x^2}=\frac{e^x(x-1)+1}{x^2} \end{aligned}$ $\left(\frac{\mathrm{e}^{\mathrm{x}}(\mathrm{x}-1)}{\mathrm{x}^2}\right)+\frac{1}{\mathrm{x}^2}$, which is a strictly increasing function.

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

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