Of `f(x)=2/(sqrt(3))tan^(-1)((2x+1)/(sqrt(3)))-log(x^2+x+1)(lambda^2-5lambda+3)x+10` is a decreasing function for all `x in R ,` find the permissible values of `lambdadot`

Explanation

`f(x)=(2) sqrt(3)(1)/(1+2x+1)sqrt(3)^(2).(2)/sqrt(3)-(2x+1)/(x^(2)+x+1)+(lambda^(2)-5lambda+3)le0`
`(-2x)/(x^(2)+x+1)+(lambda^(2)-5lambda+3)le0, forall x in R`
or `(lambda^(2)-5lambda+3le(2x))/(x^(2)+x+1),forall x I R`
Now let y =`(2x)/(x^(2)+x+1)=(2)/(x+1+(1)/(x))`
Putting x=1 and x=-1,y=`(2)/(3)`,y=-2,respectively.
So range of y in `[-2,(2)/(33)]`
From (1)`lambda^(2)-5lambda+3le-2`
or `lambda^(2)-5lambda+5le0`
or `(lambda-(5-sqrt(5)))/(2)(lambda-(5+sqrt(5)))/(2)le0`
or `lambda in [(5-sqrt(5))/(2),(5+sqrt(5))/(2)]`