Question

The range of parameter ' $b$ ' for which the function $f(x)=\underset{0}{\overset{x}{∫}}\left(b{t}^2+b+\cos t\right)dt$ is monotonic for all real values of $x$, is

• A. $[−1,1]$
• B. $(−∞,−1]∪[1,∞)$
• C. $(−∞,−1)∪(1,∞)$
• D. $(−1,1)$

Answer 1

Answer: (B)

$f(x)=\underset{0}{\overset{x}{∫}}\left(bt{t}^2+b+\cos t\right)dt$
$f^{′}(x)=b{x}^2+b+\cos x$
Case-1: $f(x)$ is monotonic increasing $∀x∈R$
$f^{′}(x)â‰\yen 0⇒b^2+b+\cos xâ‰\yen 0(\text{ minimum }\cos x=−1)$
$b{x}^2+b−1â‰\yen 0⇒b>0,D≤0$
$0−4b(b−1)≤0⇒b(b−1)â‰\yen 0⇒b∈(−∞,0]∪[1,∞)$
$b∈[1,∞)$
Case-2: $f(x)$ is monotonic decreasing $∀x∈R$
$f^{′}(x)≤0⇒b^2+b+\cos x≤0(\text{ maximum }\cos x=1)$
$b{x}^2+b+1≤0⇒b<0,D≤0$
$0−4b(b+1)≤0⇒b(b+1)â‰\yen 0⇒b∈(−∞,−1]∪[0,∞)$
$b∈(−∞,−1]$