For what values of a does the curve f(x)=x(a2-2 a-2)+ cos x is always strictly monotonic decreasing ∀ x ∈ R.

Question

For what values of a does the curve f(x)=x(a2-2 a-2)+ cos x is always strictly monotonic decreasing ∀ x ∈ R. (A) a ∈ R (B) | a | < √2 (C) 1-√2 < a < 1+√2 (D) |a| < √2-1

Tardigrade Admin · Accepted Answer

For strictly monotonic decreasing f prime(x)<0 f prime(x)=a2-2 a-2- sin x < 0 ∀ x ∈ R ⇒(a-1)2 < 3+ sin x ∀ x ∈ R ⇒(a-1)2 < 2 ⇒ 1-√2 < a < √2+1

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

$a \in R$

$∣ a ∣ < 2$

$1 - 2 < a < 1 + 2$

$∣ a ∣ < 2 - 1$

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

Q. For what values of a does the curve f(x)=x(a2−2a−2)+cosx is always strictly monotonic decreasing ∀x∈R.

a∈R

∣a∣<2​

1−2​<a<1+2​

∣a∣<2​−1