Graphical Interpretation of Differentiability

Q. Let $f\left(x\right)$ be a polynomial function satisfying the following conditions
$\underset{x\to \infty }{lim}\frac{f\left(x\right)}{|x{|}^3}=0,\underset{x\to \infty }{lim}(\sqrt{f\left(x\right)}-x)=-1andf\left(0\right)=0$
(where, [.] denotes greatest integer function) then which of the following is/are correct?

1. The number of points of discontinuity of $g\left(x\right)=\left[f\right(x\left)\right]$ in [0, 3] is 3
2. The number of points of discontinuity of $g\left(x\right)=\left[f\right(x\left)\right]$ in [0, 3] is 5
3. The number of points of non-derivability of $h\left(x\right)=|f\left(|x|\right)|$ is 4
4. The number of points of non-derivability of $h\left(x\right)=|f\left(|x|\right)|$ is 3

Q. Let $f:R\to R$ be a function defined by $f\left(x\right)=max\{x,x^3\}.$ Then the set of all points where $f$ is not differentiable, is

1. $\{-1,0\}$
2. $\{-1,0,1\}$
3. $\{0,1\}$
4. $\{-1,1\}$

Q. Let $f\left(x\right)$ be a polynomial function satisfying the following conditions
$\underset{x\to \infty }{lim}\frac{f\left(x\right)}{|x{|}^3}=0,\underset{x\to \infty }{lim}(\sqrt{f\left(x\right)}-x)=-1andf\left(0\right)=0$
(where, [.] denotes greatest integer function) then which of the following is/are correct?

1. The number of points of discontinuity of $g\left(x\right)=\left[f\right(x\left)\right]$ in [0, 3] is 3
2. The number of points of discontinuity of $g\left(x\right)=\left[f\right(x\left)\right]$ in [0, 3] is 5
3. The number of points of non-derivability of $h\left(x\right)=|f\left(|x|\right)|$ is 4
4. The number of points of non-derivability of $h\left(x\right)=|f\left(|x|\right)|$ is 3

Q. Let $S$ be the set of all points in $(-\pi ,\pi )$ at which the function, $f\left(x\right)$ = min $\{\sin x,\cos x\}$ is not differentiable. Then $S$ is a subset of which of the following?

1. $\{-\frac{\pi }{2},-\frac{\pi }{4},\frac{\pi }{4},\frac{\pi }{2}\}$
2. $\{-\frac{3\pi }{4},-\frac{\pi }{4},\frac{3\pi }{4},\frac{\pi }{4}\}$
3. $\{-\frac{3\pi }{4},-\frac{\pi }{2},\frac{\pi }{2},\frac{3\pi }{4}\}$
4. $\{-\frac{\pi }{4},0,\frac{\pi }{4}\}$

Q. Let $f:(-1,1)\to R$ be a function defined by $f\left(x\right)=max\{-|x|,-\sqrt{1-x^2}\}$. If $K$ be the set of all points at which $f$ is not differentiable, then $K$ has exactly

1. one element
2. two elements
3. three elements
4. five elements