Question

The function $f\left(x\right)=x.e^{-x}x\in R$ attains a maximum value at x =

• A. $2$
• B. $\frac{1}{e}$
• C. $1$
• D. $3$

Answer 1

The correct option is C $1$

$f\left(x\right)=x{e}^{-x}$
$f^{\prime }\left(x\right)=-x{e}^{-x}+e^{-x}=e^{-x}(1-x)$
For maxima or minima,
$f^{\prime }\left(x\right)=0$
$e^{-x}(1-x)=0$
$\Rightarrow x=1$ (since $e^{-x}\ne 0$)
$f^{\prime \prime }\left(x\right)=(x-2)e^{-x}$
$\Rightarrow f^{\prime \prime }\left(1\right)<0$
So, $f\left(x\right)$ has a maximum value at $x=1$