The functions $f(x)=x e^{-x}, \forall(x \in R)$ attains a maximum value at $x$ is equal to
The functions $f(x)=x e^{-x}, \forall(x \in R)$ attains a maximum value at $x$ is equal to
$1$
$2$
$\frac{1}{e}$
$3$
Solution
We have, $f(x)=x e^{-x}$
$
f^{\prime}(x)=-x e^{-x}+e^{-x}
$
For maximum or minimum, put $f^{\prime}(x)=0$
$
\begin{aligned}
& \Rightarrow \quad-x e^{-x}+e^{-x}=0 \Rightarrow x=1 \\
& f^{\prime \prime}(x)=x e^{-x}-e^{-x}-e^{-x}=(x-2) e^{-x} \\
& f^{\prime \prime}(1)=(1-2) e^{-1}=-\mathrm{ve}
\end{aligned}
$
$f(x)$ is maximum at $x=1$