Question

Let $f\left(x\right)$ be a non-positive continuous function and $F\left(x\right)=\underset{0}{\overset{x}{\int }}f\left(t\right)dt\forall x\ge 0$ and $f\left(x\right)\ge cF\left(x\right)$ where $c>0$ and let $g:[0,\infty )\to R$ be a function such that $\frac{dg\left(x\right)}{dx}<g\left(x\right)\forall x>0$ and $g\left(0\right)=0.$
The solution set of inequality $g\left(x\right)({\cos }^{-1}x-{\sin }^{-1}x)\le 0$

• A. $[-1,\frac{1}{\sqrt{2}}]$
• B. $[\frac{1}{\sqrt{2}},1]$
• C. $[0,\frac{1}{\sqrt{2}}]$
• D. $(0,\frac{1}{\sqrt{2}}]$

Answer 1

The correct option is A $[-1,\frac{1}{\sqrt{2}}]$

$g\left(x\right)({\cos }^{-1}x-{\sin }^{-1}x)\le 0$
or $({\cos }^{-1}x-{\sin }^{-1}x)\ge 0$ or $x\in [-1,\frac{1}{\sqrt{2}}]$