If `f(x)=e^(1-x) `then f(x) is

Options

A. increasing in `[-1//2,1]`
B. decreasing in `R`
C. increasing in `R`
D. decreasing in `[-1//2,1]`

Answer

A

Explanation

Given `f(x)=xe^(x(1-x))`
`impliesf'(x)=e^(x(1-x))+xe^(x(1-x))(1-2x)`
`=e^(x(1-x))[1+x(1-2x)]=e^(x(1-x))(1+x-2x^(2))`
`=-e^(x(1-x))(2x^(2)-x-1)`
`=-e^(x(1-x))(x-1)(2x+1)`
which is positive in `[-1/2,1]`
Therefore `f(x)` is increasing in `[-1/2,1]`