122- The figure below shows the graph of the function with equation $f(x) = -x^4 + 4x^3 + ax^2 + b$. What is $a$?
[Figure: Graph of a polynomial function with a local maximum and minimum]

• [(1)] $-18$
• [(2)] $-15$
• [(3)] $-12$
• [(4)] $-9$

118. The graph shown is the graph of the function $f(x) = 3x^4 + ax^3 + bx^2 + cx$. What is $a$?
[Figure: Graph of a function with a local minimum near $x=1$ on the positive x-axis, with the curve going to positive infinity on both sides]
(1) $-8$ (2) $-7$ (3) $-5$ (4) $-4$

123-- Point $A(-1,1)$ is a relative extremum of the function $y=x^{2}|x|+3ax^{2}+b$. The value of $\dfrac{b}{a}$ is which of the following?
(1) $-3$ (2) $-\dfrac{1}{3}$ (3) $3$ (4) $\dfrac{1}{3}$

5. Determine the polynomial function of fourth degree $y = p ( x )$ knowing that, in a Cartesian coordinate system, its graph satisfies the following conditions: －it is tangent to the $x$-axis at the origin; －it passes through the point $( 1,0 )$; －it has a stationary point at $( 2 , - 2 )$.

16. If $p ( x )$ be a polynomial of degree 3 satisfying $p ( - 1 ) = 10 , p ( 1 ) = - 6$ and $p ( x )$ has maximum at $x = - 1$ and $p ^ { \prime } ( x )$ has minima at $x = 1$. Find the distance between the local maximum and local minimum of the curve.

18. $f ( x )$ is cubic polynomial which has local maximum at $x = - 1$. If $f ( 2 ) = 18 , f ( 1 ) = - 1$ and $f ^ { \prime } ( x )$ has local minima at $x = 0$, then
(A) the distance between $( - 1,2 )$ and $( \mathrm { a } , \mathrm { f } ( \mathrm { a } ) )$, where $\mathrm { x } = \mathrm { a }$ is the point of local minima is $2 \sqrt { 5 }$
(B) $\mathrm { f } ( \mathrm { x } )$ is increasing for $\mathrm { x } \in [ 1,2 \sqrt { 5 } ]$
(C) $\mathrm { f } ( \mathrm { x } )$ has local minima at $\mathrm { x } = 1$
(D) the value of $\mathrm { f } ( 0 ) = 5$
Sol. (B), (C) The required polynomial which satisfy the condition is $\mathrm { f } ( \mathrm { x } ) = \frac { 1 } { 4 } \left( 19 \mathrm { x } ^ { 3 } - 57 \mathrm { x } + 34 \right)$ $\mathrm { f } ( \mathrm { x } )$ has local maximum at $\mathrm { x } = - 1$ and local minimum at $\mathrm { x } = 1$ [Figure]
Hence $\mathrm { f } ( \mathrm { x } )$ is increasing for $\mathrm { x } \in [ 1,2 \sqrt { 5 } ]$.

Let $p(x)$ be a polynomial of degree 4 having extremum at $x=1,2$ and $$\lim_{x\rightarrow0}\left(1+\frac{p(x)}{x^{2}}\right)=2.$$ Then the value of $p(2)$ is

Let $\mathbb { R }$ denote the set of all real numbers. For a real number $x$, let $[ x ]$ denote the greatest integer less than or equal to $x$. Let $n$ denote a natural number.
Match each entry in List-I to the correct entry in List-II and choose the correct option.

List-I

(P) The minimum value of $n$ for which the function
$$f ( x ) = \left[ \frac { 10 x ^ { 3 } - 45 x ^ { 2 } + 60 x + 35 } { n } \right]$$
is continuous on the interval $[ 1,2 ]$, is (Q) The minimum value of $n$ for which
$$g ( x ) = \left( 2 n ^ { 2 } - 13 n - 15 \right) \left( x ^ { 3 } + 3 x \right)$$
$x \in \mathbb { R }$, is an increasing function on $\mathbb { R }$, is (R) The smallest natural number $n$ which is greater than 5, such that $x = 3$ is a point of local minima of
$$h ( x ) = \left( x ^ { 2 } - 9 \right) ^ { n } \left( x ^ { 2 } + 2 x + 3 \right) ,$$
is (S) Number of $x _ { 0 } \in \mathbb { R }$ such that
$$l ( x ) = \sum _ { k = 0 } ^ { 4 } \left( \sin | x - k | + \cos \left| x - k + \frac { 1 } { 2 } \right| \right) ,$$
$x \in \mathbb { R }$, is NOT differentiable at $x _ { 0 }$, is

List-II

(1) 8
(2) 9
(3) 5
(4) 6
(5) 10

(A)	$( \mathrm { P } ) \rightarrow ( 1 )$	$( \mathrm { Q } ) \rightarrow ( 3 )$	$( \mathrm { R } ) \rightarrow ( 2 )$	$( \mathrm { S } ) \rightarrow ( 5 )$
(B)	$( \mathrm { P } ) \rightarrow ( 2 )$	$( \mathrm { Q } ) \rightarrow ( 1 )$	$( \mathrm { R } ) \rightarrow ( 4 )$	$( \mathrm { S } ) \rightarrow ( 3 )$
(C)	$( \mathrm { P } ) \rightarrow ( 5 )$	$( \mathrm { Q } ) \rightarrow ( 1 )$	$( \mathrm { R } ) \rightarrow ( 4 )$	$( \mathrm { S } ) \rightarrow ( 3 )$
(D)	$( \mathrm { P } ) \rightarrow ( 2 )$	$( \mathrm { Q } ) \rightarrow ( 3 )$	$( \mathrm { R } ) \rightarrow ( 1 )$	$( \mathrm { S } ) \rightarrow ( 5 )$

Let $a, b \in \mathbb{R}$ be such that the function $f$ given by $f(x) = \ln|x| + bx^{2} + ax$, $x \neq 0$ has extreme values at $x = -1$ and $x = 2$. Statement 1: $f$ has local maximum at $x = -1$ and at $x = 2$. Statement 2: $a = \frac{1}{2}$ and $b = \frac{-1}{4}$.
(1) Statement 1 is false, Statement 2 is true
(2) Statement 1 is true, Statement 2 is false
(3) Statement 1 is true, Statement 2 is the correct explanation for Statement 1
(4) Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation for Statement 1

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) $8 x ^ { 2 } - 6 x + 1 = 0$

Q73. 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) $8 x ^ { 2 } - 6 x + 1 = 0$

Q89. Let the set of all positive values of $\lambda$, for which the point of local minimum of the function $\left( 1 + x \left( \lambda ^ { 2 } - x ^ { 2 } \right) \right)$ satisfies $\frac { x ^ { 2 } + x + 2 } { x ^ { 2 } + 5 x + 6 } < 0$, be $( \alpha , \beta )$. Then $\alpha ^ { 2 } + \beta ^ { 2 }$ is equal to $\_\_\_\_$

Q89. Let the set of all values of $p$, for which $f ( x ) = \left( p ^ { 2 } - 6 p + 8 \right) \left( \sin ^ { 2 } 2 x - \cos ^ { 2 } 2 x \right) + 2 ( 2 - p ) x + 7$ does not have any critical point, be the interval $( a , b )$. Then $16 a b$ is equal to $\_\_\_\_$