The graph of the piecewise-defined function $f$ is shown in the figure above. The graph has a vertical tangent line at $x = - 2$ and horizontal tangent lines at $x = - 3$ and $x = - 1$. What are all values of $x , - 4 < x < 3$, at which $f$ is continuous but not differentiable?
(A) $x = 1$
(B) $x = - 2$ and $x = 0$
(C) $x = - 2$ and $x = 1$
(D) $x = 0$ and $x = 1$

Let $f$ be the function defined by $f ( x ) = \sqrt { | x - 2 | }$ for all $x$. Which of the following statements is true?
(A) $f$ is continuous but not differentiable at $x = 2$.
(B) $f$ is differentiable at $x = 2$.
(C) $f$ is not continuous at $x = 2$.
(D) $\lim _ { x \rightarrow 2 } f ( x ) \neq 0$
(E) $x = 2$ is a vertical asymptote of the graph of $f$.

In the interval $( - 2 \pi , 0 )$, the function $f ( x ) = \sin \left( \frac { 1 } { x ^ { 3 } } \right)$
(A) never changes sign
(B) changes sign only once
(C) changes sign more than once, but finitely many times
(D) changes sign infinitely many times

Let $f ( x ) = \frac { 1 - x ( 1 + | 1 - x | ) } { | 1 - x | } \cos \left( \frac { 1 } { 1 - x } \right)$ for $x \neq 1$. Then
[A] $\lim _ { x \rightarrow 1 ^ { - } } f ( x ) = 0$
[B] $\lim _ { x \rightarrow 1 ^ { - } } f ( x )$ does not exist
[C] $\lim _ { x \rightarrow 1 ^ { + } } f ( x ) = 0$
[D] $\lim _ { x \rightarrow 1 ^ { + } } f ( x )$ does not exist

Let $f : ( 0,1 ) \rightarrow \mathbb { R }$ be the function defined as $f ( x ) = [ 4 x ] \left( x - \frac { 1 } { 4 } \right) ^ { 2 } \left( x - \frac { 1 } { 2 } \right)$, where $[ x ]$ denotes the greatest integer less than or equal to $x$. Then which of the following statements is(are) true?
(A) The function $f$ is discontinuous exactly at one point in $( 0,1 )$
(B) There is exactly one point in $( 0,1 )$ at which the function $f$ is continuous but NOT differentiable
(C) The function $f$ is NOT differentiable at more than three points in $( 0,1 )$
(D) The minimum value of the function $f$ is $- \frac { 1 } { 512 }$

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ and $g : \mathbb { R } \rightarrow \mathbb { R }$ be functions defined by
$$f ( x ) = \left\{ \begin{array} { l l } x | x | \sin \left( \frac { 1 } { x } \right) , & x \neq 0 , \\ 0 , & x = 0 , \end{array} \quad \text { and } \quad g ( x ) = \begin{cases} 1 - 2 x , & 0 \leq x \leq \frac { 1 } { 2 } \\ 0 , & \text { otherwise } \end{cases} \right.$$
Let $a , b , c , d \in \mathbb { R }$. Define the function $h : \mathbb { R } \rightarrow \mathbb { R }$ by
$$h ( x ) = a f ( x ) + b \left( g ( x ) + g \left( \frac { 1 } { 2 } - x \right) \right) + c ( x - g ( x ) ) + d g ( x ) , x \in \mathbb { R }$$
Match each entry in List-I to the correct entry in List-II.
List-I
(P) If $a = 0 , b = 1 , c = 0$, and $d = 0$, then
(Q) If $a = 1 , b = 0 , c = 0$, and $d = 0$, then
(R) If $a = 0 , b = 0 , c = 1$, and $d = 0$, then
(S) If $a = 0 , b = 0 , c = 0$, and $d = 1$, then
List-II
(1) $h$ is one-one.
(2) $h$ is onto.
(3) $h$ is differentiable on $\mathbb { R }$.
(4) the range of $h$ is $[ 0,1 ]$.
(5) the range of $h$ is $\{ 0,1 \}$.
The correct option is
(A) $(\mathrm{P}) \rightarrow (4)$, $(\mathrm{Q}) \rightarrow (3)$, $(\mathrm{R}) \rightarrow (1)$, $(\mathrm{S}) \rightarrow (2)$
(B) $(\mathrm{P}) \rightarrow (5)$, $(\mathrm{Q}) \rightarrow (2)$, $(\mathrm{R}) \rightarrow (4)$, $(\mathrm{S}) \rightarrow (3)$
(C) $(\mathrm{P}) \rightarrow (5)$, $(\mathrm{Q}) \rightarrow (3)$, $(\mathrm{R}) \rightarrow (2)$, $(\mathrm{S}) \rightarrow (4)$
(D) $(\mathrm{P}) \rightarrow (4)$, $(\mathrm{Q}) \rightarrow (2)$, $(\mathrm{R}) \rightarrow (1)$, $(\mathrm{S}) \rightarrow (3)$

Let $f : R \rightarrow R$ be a function defined by $f ( x ) = \operatorname { Min } \{ x + 1 , | x | + 1 \}$. Then which of the following is true?
(1) $f ( x ) \geq 1$ for all $x \in R$
(2) $f ( x )$ is not differentiable at $x = 1$
(3) $f ( x )$ is differentiable everywhere
(4) $f ( x )$ is not differentiable at $x = 0$

Consider the function $f : R \rightarrow R$ defined by $f ( x ) = \left\{ \begin{array} { c c } \left( 2 - \sin \left( \frac { 1 } { x } \right) \right) | x | , & x \neq 0 \\ 0 , & x = 0 \end{array} \right.$. Then $f$ is:
(1) monotonic on $( - \infty , 0 ) \cup ( 0 , \infty )$
(2) not monotonic on $( - \infty , 0 )$ and $( 0 , \infty )$
(3) monotonic on $( 0 , \infty )$ only
(4) monotonic on $( - \infty , 0 )$ only

Let $\mathrm { f } ( \mathrm { x } ) = \min \left\{ \sqrt { 2 } \mathrm { x } , \mathrm { x } ^ { 2 } \right\}$ and $\mathrm { g } ( \mathrm { x } ) = | x | \left[ x ^ { 2 } \right ]$
If $x \in ( - 2,2 )$ then sum of all values of $f ( x )$ at those $x$ values where $g ( x )$ is non-differentiable ([.] denotes GIF). (A) $2 - \sqrt { 3 }$ (B) $1 / - \sqrt { 3 }$ (C) [answer] (D) $2 - \sqrt { 2 }$