Question

Find the intervals of monotonicity for the following function.

$f\left(x\right)=\frac{e^x}{x}$

• A. I in $(2,\infty )$ & D in $(-\infty ,0)\cup (0,2)$
• B. I in $(3,\infty )$ & D in $(-\infty ,0)\cup (0,3)$
• C. I in $(1,\infty )$ & D in $(-\infty ,0)\cup (0,1)$
• D. I in $(3,\infty )$ & D in $(-\infty ,1)\cup (1,3)$

Answer 1

Answer: (C)

I in $(1,\infty )$ & D in $(-\infty ,0)\cup (0,1)$

Given, $f\left(x\right)=\frac{e^x}{x}$

${lim}_{x\to 0^{+}}$ $=\frac{e^x-1}{x}+\frac{1}{x}$$=\infty$

Similarly, ${lim}_{x\to 0^{-}}$ $=\frac{e^x-1}{x}+\frac{1}{x}$ $=-\infty$

Hence discontinuous at $x=0$.

Now $f^{\prime }\left(x\right)>0$ implies $x\left(e^x\right)-e^x>0$

$\Rightarrow e^x(x-1)>0$

$\Rightarrow x>1$

Hence, increasing for $(1,\infty )$ and decreasing for $(-\infty ,1)-\left\{0\right\}$