Geometrical Applications of Differential Equations

Q. 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 total number of root(s) of the equation $f\left(x\right)=g\left(x\right)$ is/are

1. $\infty$
2. $1$
3. $2$
4. $0$

Q. 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 $g\left(x\right)({\cos }^{-1}x-{\sin }^{-1}x)\le 0$

1. $[-1,\frac{1}{\sqrt{2}}]$
2. $[\frac{1}{\sqrt{2}},1]$
3. $[0,\frac{1}{\sqrt{2}}]$
4. $(0,\frac{1}{\sqrt{2}}]$