Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be defined as $$f(x) = \begin{cases} \left[e^x\right], & x < 0 \\ ae^x + [x-1], & 0 \leq x < 1 \\ b + [\sin(\pi x)], & 1 \leq x < 2 \\ \left[e^{-x}\right] - c, & x \geq 2 \end{cases}$$ where $a, b, c \in \mathbb{R}$ and $[t]$ denotes greatest integer less than or equal to $t$. Then, which of the following statements is true?
(1) There exists $a, b, c \in \mathbb{R}$ such that $f$ is continuous
(2) If $f$ is discontinuous at exactly one point, then $a + b + c = 1$
(3) If $f$ is discontinuous at exactly one point, then $a + b + c \neq 1$
(4) $f$ is discontinuous at at least two points, for any values of $a, b, c \in \mathbb{R}$

Let $f(x) = \begin{cases} \left\lfloor 4x ^ { 2 } - 8x + 5 \right\rfloor, & \text{if } 8x ^ { 2 } - 6x + 1 \geq 0 \\ \left\lfloor 4x ^ { 2 } - 8x + 5 \right\rfloor, & \text{if } 8x ^ { 2 } - 6x + 1 < 0 \end{cases}$, where $\lfloor \alpha \rfloor$ denotes the greatest integer less than or equal to $\alpha$. Then the number of points in $R$ where $f$ is not differentiable is $\_\_\_\_$.

Let $[ x ]$ be the greatest integer $\leq x$. Then the number of points in the interval $( - 2,1 )$ where the function $f ( x ) = | [ x ] | + \sqrt { x - [ x ] }$ is discontinuous, is $\_\_\_\_$ .

Consider the function $\mathrm { f } : ( 0,2 ) \rightarrow \mathrm { R }$ defined by $\mathrm { f } ( \mathrm { x } ) = \frac { \mathrm { x } } { 2 } + \frac { 2 } { \mathrm { x } }$ and the function $\mathrm { g } ( \mathrm { x } )$ defined by $\mathrm { gx } = \begin{array} { c c } \min \{ \mathrm { f } ( \mathrm { t } ) \} , & 0 < \mathrm { t } \leq \mathrm { x } \text { and } 0 < \mathrm { x } \leq 1 \\ \frac { 3 } { 2 } + \mathrm { x } , & 1 < \mathrm { x } < 2 \end{array}$. Then
(1) g is continuous but not differentiable at $\mathrm { x } = 1$
(2) g is not continuous for all $\mathrm { x } \in ( 0,2 )$
(3) g is neither continuous nor differentiable at $\mathrm { x } = 1$
(4) $g$ is continuous and differentiable for all $x \in ( 0,2 )$

Let $f : [ - 1,2 ] \rightarrow \mathbf { R }$ be given by $f ( x ) = 2 x ^ { 2 } + x + \left[ x ^ { 2 } \right] - [ x ]$, where $[ t ]$ denotes the greatest integer less than or equal to $t$. The number of points, where $f$ is not continuous, is :
(1) 5
(2) 6
(3) 3
(4) 4

Let $[ \mathrm { t } ]$ denote the greatest integer less than or equal to t. Let $f : [ 0 , \infty ) \rightarrow \mathbf { R }$ be a function defined by $f ( x ) = \left[ \frac { x } { 2 } + 3 \right] - [ \sqrt { x } ]$. Let $S$ be the set of all points in the interval $[ 0,8 ]$ at which $f$ is not continuous. Then $\sum _ { \mathrm { a } \in \mathrm { S } } \mathrm { a }$ is equal to $\_\_\_\_$

Let $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function $f(x) = [x] + |x - 2|$, $-2 < x < 3$, is not continuous and not differentiable. Then $\mathrm{m} + \mathrm{n}$ is equal to:
(1) 6
(2) 8
(3) 9
(4) 7

Let $\mathrm { f } ( x ) = \left\{ \begin{array} { l l } 3 x , & x < 0 \\ \min \{ 1 + x + [ x ] , x + 2 [ x ] \} , & 0 \leq x \leq 2 \\ 5 , & x > 2 , \end{array} \right.$ where [.] denotes greatest integer function. If $\alpha$ and $\beta$ are the number of points, where f is not continuous and is not differentiable, respectively, then $\alpha + \beta$ equals

Q72. Let $f : [ - 1,2 ] \rightarrow \mathbf { R }$ be given by $f ( x ) = 2 x ^ { 2 } + x + \left[ x ^ { 2 } \right] - [ x ]$, where $[ t ]$ denotes the greatest integer less than or equal to $t$. The number of points, where $f$ is not continuous, is :
(1) 5
(2) 6
(3) 3
(4) 4

Q86. Let $[ \mathrm { t } ]$ denote the greatest integer less than or equal to t . Let $f : [ 0 , \infty ) \rightarrow \mathbf { R }$ be a function defined by $f ( x ) = \left[ \frac { x } { 2 } + 3 \right] - [ \sqrt { x } ]$. Let $S$ be the set of all points in the interval $[ 0,8 ]$ at which $f$ is not continuous. Then $\sum _ { \mathrm { a } \in \mathrm { S } } \mathrm { a }$ is equal to $\_\_\_\_$

Consider the function $f ( x ) = \left\{ \begin{array} { l l l } 8 e ^ { 2 x - 4 } & \text { if } & x \leq 2 \\ \frac { x ^ { 3 } - 4 x } { x - 2 } & \text { if } & x > 2 \end{array} \right.$ and it is requested:
a) (0.75 points) Study the continuity of $f$ at $x = 2$.
b) (1 point) Calculate the asymptotes of $f ( x )$. Is there any vertical asymptote?
c) (0.75 points) Calculate $\int _ { 0 } ^ { 2 } f ( x ) d x$

Given the function $f(x) = \left\{ \begin{array}{lll} \frac{x-1}{x^{2}-1} & \text{if} & x < 1, x \neq -1 \\ \frac{x^{2}+1}{4x} & \text{if} & x \geq 1 \end{array} \right.$, find:\ a) (0.5 points) Calculate $f(0)$ and $(f \circ f)(0)$.\ b) (1.25 points) Study the continuity and differentiability of $f(x)$ at $x = 1$ and determine if there exists a relative extremum at that point.\ c) (0.75 points) Study its asymptotes.

Let the function
$$f ( x ) = \left\{ \begin{array} { l l l } ( x - 1 ) ^ { 2 } & \text { if } & x \leq 1 \\ ( x - 1 ) ^ { 3 } & \text { if } & x > 1 \end{array} \right.$$
a) (0.5 points) Study its continuity on $[ - 4 ; 4 ]$.\ b) (1 point) Analyze its differentiability and growth on [-4;4].\ c) (1 point) Determine whether the function $g ( x ) = f ^ { \prime } ( x )$ is defined, continuous and differentiable at $x = 1$.

Let the function
$$f ( x ) = \left\{ \begin{array} { l l l } \frac { 2 x + 1 } { x } & \text { if } & x < 0 \\ x ^ { 2 } - 4 x + 3 & \text { if } & x \geq 0 \end{array} \right.$$
a) ( 0.75 points) Study the continuity of $f ( x )$ in $\mathbb { R }$. b) ( 0.25 points) Is $f ( x )$ differentiable at $x = 0$ ? Justify your answer. c) ( 0.75 points) Calculate, if they exist, the equations of its horizontal and vertical asymptotes. d) ( 0.75 points) Determine for $x \in ( 0 , \infty )$ the point on the graph of $f ( x )$ where the slope of the tangent line is zero and obtain the equation of the tangent line at that point. At the point obtained, does $f ( x )$ achieve any relative extremum? If so, classify it.

Given the real function of a real variable defined on its domain as $f ( x ) = \left\{ \begin{array} { l l l } \frac { x ^ { 2 } } { 2 + x ^ { 2 } } & \text { if } & x \leq - 1 \\ \frac { 2 x ^ { 2 } } { 3 - 3 x } & \text { if } & x > - 1 \end{array} \right.$, find:\ a) ( 0.75 points) Study the continuity of the function on $\mathbb{R}$.\ b) (1 point) Calculate the following limit: $\lim _ { x \rightarrow - \infty } f ( x ) ^ { 2 x ^ { 2 } - 1 }$.\ c) (0.75 points) Calculate the following integral: $\int _ { - 1 } ^ { 0 } f ( x ) d x$.

Let the function $f ( x ) = \left\{ \begin{array} { l l l } x ^ { 2 } - 6 x + 11 & \text { if } & x < 2 \\ \sqrt { 5 x - 1 } & \text { if } & x \geq 2 \end{array} \right.$. a) ( 0.5 points) Study the continuity of the function in $\mathbb { R }$. b) (1 point) Study the relative extrema of the function in the interval ( 1,3 ). c) (1 point) Calculate the area enclosed by the function and the $x$-axis between $x = 1$ and $x = 3$.