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

Options

A. incresing on [-1/2,1]
B. decresing on R
C. increcasing on R
D. decreasing on [-1/2 ,1]

Answer

A

Explanation

We have
`f(x)=xe^(x(1-x))`
`rArr f(x)=e^(x(1-x))+x(1-2x)e^(x(1-2))`
`f''(x)=(1+x-2x^2)e^(x(1-x))`
`rArr f(x)=-(x-1)(2x +1)e^(x(1-x))`
Since `e^(x(1-x)) gt 0` for all x .Therefore ,sign of f (x) for different values of x are as shown in

Clearly f(x) is incresing on [-1/2,1] and decreasing on `(-oo,-1//2 ] cup [1,oo)`