If $f\left(x\right)={\int }_0^x \left[(a+1)(t+1{)}^2-(a-1)\left(t^2+t+1\right)\right]dt$ then a possible positive value of ' a ', for which $f^{\text{'}}\left(x\right)=0$ has equal roots, is

$f\left(x\right)={\int }_0^x \left[(a+1)(t+1{)}^2-(a-1)\left(t^2+t+1\right)\right]dt$

diff with respect to x

$f^{\text{'}}\left(x\right)= \left[(a+1)(x+1{)}^2-(a-1)\left(x^2+x+1\right)\right]$

$f^{\text{'}}\left(x\right)= \left[(a+1)(x^2+2x+1)-(a-1)\left(x^2+x+1\right)\right]$

$f^{\text{'}}\left(x\right)=\left[2x^2+\left(3+a\right)x+2\right]$

Discriminant=0

a=-7,1