If $f$ \& $g$ are differentiable functions in $[ 0,1 ]$ satisfying $f ( 0 ) = 2 = g ( 1 ) , g ( 0 ) = 0$ \& $f ( 1 ) = 6$, then for some $c \in ] 0,1 [$
(1) $f ^ { \prime } ( c ) = g ^ { \prime } ( c )$
(2) $f ^ { \prime } ( c ) = 2 g ^ { \prime } ( c )$
(3) $2 f ^ { \prime } ( c ) = g ^ { \prime } ( c )$
(4) $2 f ^ { \prime } ( c ) = 3 g ^ { \prime } ( c )$

If $x = - 1$ and $x = 2$ are extreme points of $f ( x ) = \alpha \log | x | + \beta x ^ { 2 } + x$, then
(1) $\alpha = 2 , \beta = - \frac { 1 } { 2 }$
(2) $\alpha = 2 , \beta = \frac { 1 } { 2 }$
(3) $\alpha = - 6 , \beta = \frac { 1 } { 2 }$
(4) $\alpha = - 6 , \beta = - \frac { 1 } { 2 }$

If the Rolle's theorem holds for the function $f ( x ) = 2 x ^ { 3 } + a x ^ { 2 } + b x$ in the interval $[ - 1,1 ]$ for the point $c = \frac { 1 } { 2 }$, then the value of $2 a + b$ is:
(1) $-1$
(2) 2
(3) 1
(4) $-2$

Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = \begin{cases} k - 2x, & \text{if } x \leq -1 \\ 2x + 3, & \text{if } x > -1 \end{cases}$. If $f$ has a local minimum at $x = -1$, then a possible value of $k$ is:
(1) $0$
(2) $-\frac{1}{2}$
(3) $-1$
(4) $1$

Let $f ( x )$ be a polynomial of degree four and having its extreme values at $x = 1$ and $x = 2$. If $\lim _ { x \rightarrow 0 } \left[ 1 + \frac { f ( x ) } { x ^ { 2 } } \right] = 3$, then $f ( 2 )$ is equal to
(1) 4
(2) - 8
(3) - 4
(4) 0

If the function $f$ given by $f ( x ) = x ^ { 3 } - 3 ( a - 2 ) x ^ { 2 } + 3 a x + 7$, for some $a \in R$ is increasing in $( 0,1 ]$ and decreasing in $[ 1,5 )$, then a root of the equation, $\frac { f ( x ) - 14 } { ( x - 1 ) ^ { 2 } } = 0 , ( x \neq 1 )$ is :
(1) 7
(2) - 7
(3) 6
(4) 5

If $p(x)$ be a polynomial of degree three that has a local maximum value 8 at $x = 1$ and a local minimum value 4 at $x = 2$ then $p(0)$ is equal to
(1) 6
(2) $-12$
(3) 24
(4) 12

If Rolle's theorem holds for the function $f ( x ) = x ^ { 3 } - a x ^ { 2 } + b x - 4 , x \in [ 1,2 ]$ with $f ^ { \prime } \left( \frac { 4 } { 3 } \right) = 0$, then ordered pair $( a , b )$ is equal to :
(1) $( - 5 , - 8 )$
(2) $( - 5,8 )$
(3) $( 5,8 )$
(4) $( 5 , - 8 )$

Let $f ( x )$ be a polynomial of degree 6 in $x$, in which the coefficient of $x ^ { 6 }$ is unity and it has extrema at $x = - 1$ and $x = 1$. If $\lim _ { x \rightarrow 0 } \frac { f ( x ) } { x ^ { 3 } } = 1$, then $5 \cdot f ( 2 )$ is equal to

The range of $a \in R$ for which the function $f ( x ) = ( 4 a - 3 ) \left( x + \log _ { e } 5 \right) + 2 ( a - 7 ) \cot \left( \frac { x } { 2 } \right) \sin ^ { 2 } \left( \frac { x } { 2 } \right) , x \neq 2 n \pi , n \in N$, has critical points, is :
(1) $( - 3,1 )$
(2) $\left[ - \frac { 4 } { 3 } , 2 \right]$
(3) $[ 1 , \infty )$
(4) $( - \infty , - 1 ]$

Let the function $f ( x ) = 2 x ^ { 3 } + ( 2 p - 7 ) x ^ { 2 } + 3 ( 2 p - 9 ) x - 6$ have a maxima for some value of $x < 0$ and a minima for some value of $x > 0$. Then, the set of all values of $p$ is
(1) $\left( \frac { 9 } { 2 } , \infty \right)$
(2) $\left( 0 , \frac { 9 } { 2 } \right)$
(3) $\left( - \infty , \frac { 9 } { 2 } \right)$
(4) $\left( - \frac { 9 } { 2 } , \frac { 9 } { 2 } \right)$

If the functions $f(x) = \frac{x^{3}}{3} + 2bx + \frac{ax^{2}}{2}$ and $g(x) = \frac{x^{3}}{3} + ax + bx^{2}$, $a \neq 2b$ have a common extreme point, then $a + 2b + 7$ is equal to
(1) 4
(2) $\frac{3}{2}$
(3) 3
(4) 6

If the function $f ( x ) = 2 x ^ { 3 } - 9 x ^ { 2 } + 12 \mathrm { a } ^ { 2 } x + 1 , \mathrm { a } > 0$ has a local maximum at $x = \alpha$ and a local minimum at $x = \alpha ^ { 2 }$, then $\alpha$ and $\alpha ^ { 2 }$ are the roots of the equation : (1) $x ^ { 2 } - 6 x + 8 = 0$ (2) $x ^ { 2 } + 6 x + 8 = 0$ (3) $8 x ^ { 2 } + 6 x - 1 = 0$ (4) $8 x ^ { 2 } - 6 x + 1 = 0$

A third-degree real-coefficient polynomial function $P(x)$ with leading coefficient 1 has two of its roots as $-5$ and $2$.
If $P(x)$ has a local extremum at the point $x = 0$, what is the third root?
A) $\frac { 1 } { 2 }$
B) $\frac { 3 } { 2 }$
C) $\frac { 7 } { 3 }$
D) $\frac { -5 } { 2 }$
E) $\frac { -10 } { 3 }$

Let $a$ and $b$ be real numbers. It is known that the polynomial
$$f ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + 1$$
is

• increasing on the interval $( - \infty , 1 )$,
• decreasing on the interval $( 1,5 )$,
• increasing on the interval $( 5 , \infty )$.

Accordingly, what is $f ( 2 )$?
A) 0
B) 3
C) 6
D) 9
E) 12

Let $k$ and $m$ be real numbers. The functions $f$ and $g$ defined on the set of real numbers are
$$\begin{aligned} & f(x) = 2x^{3} - 9x^{2} - mx - k \\ & g(x) = x^{3} \cdot f(x) \end{aligned}$$
The functions $f$ and $g$ have local extrema at $x = -1$.
What is the sum $k + m$?
A) 31 B) 33 C) 35 D) 37 E) 39