The curve `y =xe^(x)` has minimum value equal to

Options

A. `-(1)/(e )`
B. `(1)/(e )`
C. `-e`
D. e

Answer

A

Explanation

Let `y=xe^(x)`
Differntiate both side w.r.t 'x'
`rarr (dy)/(dx)=e^(x)+xe^(x)=e^(x)(1+x)`
put `(dy)/(dx)=0`
`rarr e^(x)(1+x)=0 rarr x=-1`
Now `(d^(2(y))/(dx^(2))=e^(x)+e^(x)(1+x)=e^(x)(x+2)`
`(d^(2))/(dx^(2))_(x=-1)=(1)/(e )+0 gt 0`
Hence `y=xe^(x)` is minimum funciton and `y_(min)=-(1)/(e )`