Question

For what values of a does the curve $f(x)=x\left(a^2-2 a-2\right)+\cos x$ is always strictly monotonic decreasing $\forall x \in R$.

• A. $a \in R$
• B. $| a | < \sqrt{2}$
• C. $1-\sqrt{2} < a < 1+\sqrt{2}$
• D. $|a| < \sqrt{2}-1$

Answer 1

Answer: (C)

For strictly monotonic decreasing $f^{\prime}(x)<0$
$f^{\prime}(x)=a^2-2 a-2-\sin x < 0 \forall x \in R$
$\Rightarrow(a-1)^2 < 3+\sin x \,\, \forall x \in R$
$\Rightarrow(a-1)^2 < 2 $
$ \Rightarrow 1-\sqrt{2} < a < \sqrt{2}+1$