Question

The curve $y = xe^x$ has minimum value equal to

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

Answer 1

Answer: (A)

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