Question

The curve $y=x{e}^x$ has minimum value equal to

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

Answer 1

Answer: (A)

Let $y=x{e}^x$.
Differentiate both side w.r.t. ‘$x$’.
$\Rightarrow \frac{dy}{dx}=e^x+x{e}^x=e^x\left(1+x\right)$
Put $\frac{dy}{dx}=0$
$\Rightarrow e^x\left(1+x\right)=0$
$\Rightarrow x=-1$
Now, $\frac{d^{2}y}{d{x}^2}=e^x+e^x\left(1+x\right)=e^x\left(x+2\right)$
${\left(\frac{d^{2}y}{d{x}^2}\right)}_{\left(x=-1\right)}=\frac{1}{e}+0>0$
Hence, $y=x{e}^x$ is minimum function and
$y_{\min }=-\frac{1}{e}$