The range of parameter ‘b’ for which the function $f\left(x\right)=\underset{0}{\overset{x}{\int }}\left(b{t}^2+b+\cos t\right)dt$ is

$f\text{'}\left(x\right)=b{x}^2+b+\cos x$

$f\left(x\right)$ is entirely strictly increasing, if $f\text{'}\left(x\right)>0,\forall x\in R$

$\Rightarrow b{x}^2+b+\cos x>0,\forall x\in R\Rightarrow b{x}^2+b-1>0,\forall x\in R$

$\Rightarrow b>0\&\Delta <0\Rightarrow b>0\&0-4b\left(b-1\right)<0$

$\Rightarrow b>0\&b\left(b-1\right)>0\Rightarrow b>1$

$f\left(x\right)$ is entirely strictly decreasing, if $f\text{'}\left(x\right)<0,\forall x\in R$

$\Rightarrow b{x}^2+b+\cos x<0,\forall x\in R\Rightarrow b{x}^2+b+1<0,\forall x\in R$

$\Rightarrow b<0\&\Delta <0\Rightarrow b<0\&0-4b\left(b+1\right)<0$

$\Rightarrow b<0\&b\left(b+1\right)>0\Rightarrow b<-1$