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Q.
The range of parameter ' $b$ ' for which the function $f(x)=\int\limits_0^x\left(b t^2+b+\cos t\right) d t$ is monotonic for all real values of $x$, is
Application of Derivatives
Solution:
$f ( x )=\int\limits_0^{ x }\left( bt t ^2+ b +\cos t \right) dt$
$f^{\prime}(x)=b x^2+b+\cos x$
Case-1: $f ( x )$ is monotonic increasing $\forall x \in R$
$f^{\prime}(x) \geq 0 \Rightarrow b^2+b+\cos x \geq 0(\text { minimum } \cos x=-1) $
$b x^2+b-1 \geq 0 \Rightarrow b>0, D \leq 0$
$0-4 b(b-1) \leq 0 \Rightarrow b(b-1) \geq 0 \Rightarrow b \in(-\infty, 0] \cup[1, \infty) $
$b \in[1, \infty)$
Case-2: $f ( x )$ is monotonic decreasing $\forall x \in R$
$f^{\prime}(x) \leq 0 \Rightarrow b^2+b+\cos x \leq 0(\text { maximum } \cos x=1) $
$b x^2+b+1 \leq 0 \Rightarrow b<0, D \leq 0 $
$0-4 b(b+1) \leq 0 \Rightarrow b(b+1) \geq 0 \Rightarrow b \in(-\infty,-1] \cup[0, \infty) $
$b \in(-\infty,-1]$