Question

Which of the following statements is (are) INCORRECT about $f\left(x\right)=\frac{e^x}{x}$?

• A. $f\left(x\right)$ is increasing in $(0,\infty )$.
• B. Tangent on the curve $y=f\left(x\right)$ at $x=1$ is perpendicular to $x-$axis.
• C. Tangent on the curve $y=f\left(x\right)$ at $x=1$ is parallel to $x-$axis.
• D. Range of $f\left(x\right)$ is $[e,\infty )$.

Answer 1

Answer: (A), (B), (D)

Range of $f\left(x\right)$ is $[e,\infty )$.

$f\left(x\right)=\frac{e^x}{x}$
Domain of $f\left(x\right)$ is $(-\infty ,0)\cup (0,\infty )$
$f^{\prime }\left(x\right)=\frac{e^x(x-1)}{x^2}$


At $x=1,$ $f^{\prime }\left(x\right)=0$
$\therefore$ Tangent on the curve $y=f\left(x\right)$ at $x=1$ is parallel to $x-$axis.


For $x<0,$ we have
$\underset{x\to 0^{-}}{lim}f\left(x\right)=-\infty$ and $\underset{x\to -\infty }{lim}f\left(x\right)=0$
For $x>0,$ we have
Minimum value of $f\left(x\right)$ is $f\left(1\right)=e$
$\underset{x\to 0^{+}}{lim}f\left(x\right)=\infty$ and $\underset{x\to \infty }{lim}f\left(x\right)=\infty$
$\therefore$ Range of $f\left(x\right)$ is $(-\infty ,0)\cup [e,\infty )$