Draw the graph of `f(x)=(x^2-x+1)/(x^2+x+1)` .

Answer

Tow distnict roots for k `in (0,1//e)`
No real roots for `klt0`
One real root for k=0 or `k ge 1//e`

Explanation

The given function is continous for all `x in R`.
`f(x) =2(x^(2-1))/(x^(2)+x+1)^(2)`
`f(x)=0rarrx=pm1`
Sign scheme of f(X) is as follows

From the sign scheme x=1 is point of minima and x =-1 is point of maxima .
Also f(1)=`1/3 and f(-1)=3f(0)=1`
further `underset(xrarr pm oo)lim (x^(2)-x+1)/(x^(1)+x+1)=1`
Now mark points (-1,3) and `(1,1//3)` on the coordinate axis
In (-1,1),f(x) decrese
In (1,oo),f(x) increases and approaches to '1' at infinity
In `(-oo,-1),f(x) `increasese from '1' at negative infinityt to 3 at x=-1
From the above information graph of y = f(x) is shown in the following figure