Question

For the function, find the intervals where it is increasing and decreasing, and JUSTIFY your conclusion. Construct a sign chart to help you organize the information, but do not use a calculator. f(x) = xex 1

Answer 1

First we have to derivate the function

$\begin{gathered} f(x)=xe^{\frac{1}{x}}\rightarrow \\ f^{\prime}(x)=e^{\frac{1}{x}}+x\cdot(e^{\frac{1}{x}}\cdot\frac{-1}{x^2})=e^{\frac{1}{x}}(1-\frac{1}{x}) \end{gathered}$

We have to find when the derivate is 0

$e^{\frac{1}{x}}(1-\frac{1}{x})=0\rightarrow1-\frac{1}{x}=0\rightarrow x=1$

we have to consider x=0, because we can not divide by 0

so for a number lower than 0 we get that

$f^{\prime}(x)>0$

so it is increasing

for a number between 0 and 1 we get

$f^{\prime}(x)<0$

so it is decreasing

for a number greater than 1 we get

$f^{\prime}(x)>0$

it is increasing