A cubic polynomial function $f$ is defined by $$f(x) = 4x^{3} + ax^{2} + bx + k$$ where $a$, $b$, and $k$ are constants. The function $f$ has a local minimum at $x = -1$, and the graph of $f$ has a point of inflection at $x = -2$.
(a) Find the values of $a$ and $b$.
(b) If $\displaystyle\int_{0}^{1} f(x)\,dx = 32$, what is the value of $k$?

Let $f$ be the function defined by $f(x) = k\sqrt{x} - \ln x$ for $x > 0$, where $k$ is a positive constant.
(a) Find $f^{\prime}(x)$ and $f^{\prime\prime}(x)$.
(b) For what value of the constant $k$ does $f$ have a critical point at $x = 1$? For this value of $k$, determine whether $f$ has a relative minimum, relative maximum, or neither at $x = 1$. Justify your answer.
(c) For a certain value of the constant $k$, the graph of $f$ has a point of inflection on the $x$-axis. Find this value of $k$.

Let $g$ be the function given by $g ( x ) = x ^ { 2 } e ^ { k x }$, where $k$ is a constant. For what value of $k$ does $g$ have a critical point at $x = \frac { 2 } { 3 }$ ?
(A) $-3$
(B) $- \frac { 3 } { 2 }$
(C) $- \frac { 1 } { 3 }$
(D) 0
(E) There is no such $k$.

In the plane with an orthonormal coordinate system ($\mathrm { O } ; \vec { \imath } , \vec { \jmath }$), the representative curve $\mathscr { C }$ of a function $f$ defined and differentiable on the interval $] 0 ; + \infty [$ is given.
We have the following information:

• the points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ have coordinates $(1,0)$, $(1,2)$, $(0,2)$ respectively;
• the curve $\mathscr { C }$ passes through point B and the line (BC) is tangent to $\mathscr { C }$ at B;
• there exist two positive real numbers $a$ and $b$ such that for every strictly positive real number $x$, $$f ( x ) = \frac { a + b \ln x } { x } .$$

1. 1. [a.] Using the graph, give the values of $f ( 1 )$ and $f ^ { \prime } ( 1 )$.
   2. [b.] Verify that for every strictly positive real number $x , f ^ { \prime } ( x ) = \frac { ( b - a ) - b \ln x } { x ^ { 2 } }$.
   3. [c.] Deduce the real numbers $a$ and $b$.
2. 1. [a.] Justify that for every real number $x$ in the interval $] 0 , + \infty [$, $f ^ { \prime } ( x )$ has the same sign as $- \ln x$.
   2. [b.] Determine the limits of $f$ at 0 and at $+ \infty$. We may note that for every strictly positive real number $x$, $f ( x ) = \frac { 2 } { x } + 2 \frac { \ln x } { x }$.
   3. [c.] Deduce the table of variations of the function $f$.
3. 1. [a.] Prove that the equation $f ( x ) = 1$ has a unique solution $\alpha$ on the interval $] 0,1 ]$.
   2. [b.] By similar reasoning, we prove that there exists a unique real number $\beta$ in the interval $] 1 , + \infty [$ such that $f ( \beta ) = 1$. Determine the integer $n$ such that $n < \beta < n + 1$.
4. The following algorithm is given.
   

   	\begin{tabular}{l} Variables:
   $a , b$ and $m$ are real numbers.
   Initialization:
   Assign to $a$ the value 0.
   Assign to $b$ the value 1.
   Processing:
   While $b - a > 0.1$
   Assign to $m$ the value $\frac { 1 } { 2 } ( a + b )$.
   If $f ( m ) < 1$ then Assign to $a$ the value $m$. Otherwise Assign to $b$ the value $m$.
   End If.
   End While.
   Output:
   Display $a$.
   Display $b$.

   \hline \end{tabular}
   
   1. [a.] Run this algorithm by completing the table below, which you will copy onto your answer sheet.
      

      	step 1	step 2	step 3	step 4	step 5
      $a$	0
      $b$	1
      $b - a$
      $m$

      
      
   2. [b.] What do the values displayed by this algorithm represent?
   3. [c.] Modify the algorithm above so that it displays the two bounds of an interval containing $\beta$ with amplitude $10 ^ { - 1 }$.
5. The purpose of this question is to prove that the curve $\mathscr { C }$ divides the rectangle OABC into two regions of equal area.
   1. [a.] Justify that this amounts to proving that $\int _ { \frac { 1 } { \mathrm { e } } } ^ { 1 } f ( x ) \mathrm { d } x = 1$.
   2. [b.] By noting that the expression of $f ( x )$ can be written as $\frac { 2 } { x } + 2 \times \frac { 1 } { x } \times \ln x$, complete the proof.

When the local minimum value of the function $f ( x ) = ( x - 1 ) ^ { 2 } ( x - 4 ) + a$ is 10, find the value of the constant $a$. [3 points]

On the coordinate plane, for a cubic function $f ( x ) = x ^ { 3 } + a x ^ { 2 } + b x$ and a real number $t$, let P be the point where the tangent line to the curve $y = f ( x )$ at the point $( t , f ( t ) )$ intersects the $y$-axis. Let $g ( t )$ be the distance from the origin to point P. The function $f ( x )$ and the function $g ( t )$ satisfy the following conditions. (가) $f ( 1 ) = 2$ (나) The function $g ( t )$ is differentiable on the entire set of real numbers. What is the value of $f ( 3 )$? (Here, $a , b$ are constants.) [4 points]
(1) 21
(2) 24
(3) 27
(4) 30
(5) 33

The function $f ( x ) = 2 x ^ { 3 } - 12 x ^ { 2 } + a x - 4$ has a local maximum value $M$ at $x = 1$. Find the value of $a + M$. (Here, $a$ is a constant.) [3 points]

Two polynomial functions $f ( x )$ and $g ( x )$ satisfy $$g ( x ) = \left( x ^ { 3 } + 2 \right) f ( x )$$ for all real numbers $x$. If $g ( x )$ has a local minimum value of 24 at $x = 1$, find the value of $f ( 1 ) - f ^ { \prime } ( 1 )$. [4 points]

For all cubic functions $f ( x )$ satisfying $f ( 0 ) = 0$ and the following conditions, let $M$ be the maximum value and $m$ be the minimum value of $\frac { f ^ { \prime } ( 0 ) } { f ( 0 ) }$. What is the value of $M m$? [4 points] (가) The function $| f ( x ) |$ is not differentiable only at $x = - 1$. (나) The equation $f ( x ) = 0$ has at least one real root in the closed interval $[ 3,5 ]$.
(1) $\frac { 1 } { 15 }$
(2) $\frac { 1 } { 10 }$
(3) $\frac { 2 } { 15 }$
(4) $\frac { 1 } { 6 }$
(5) $\frac { 1 } { 5 }$

A cubic function $f ( x )$ with leading coefficient 1 and $f ( 1 ) = 0$ satisfies $$\lim _ { x \rightarrow 2 } \frac { f ( x ) } { ( x - 2 ) \left\{ f ^ { \prime } ( x ) \right\} ^ { 2 } } = \frac { 1 } { 4 }$$ Find the value of $f ( 3 )$. [4 points]
(1) 4
(2) 6
(3) 8
(4) 10
(5) 12

For the function $f ( x ) = x ^ { 3 } - 3 x + a$, when the local maximum value is 7, what is the value of the constant $a$?
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

The function $f ( x ) = - x ^ { 4 } + 8 a ^ { 2 } x ^ { 2 } - 1$ has local maxima at $x = b$ and $x = 2 - 2 b$. What is the value of $a + b$? (Note: $a , b$ are constants with $a > 0 , b > 1$.) [3 points]
(1) 3
(2) 5
(3) 7
(4) 9
(5) 11

A cubic function $f ( x )$ with positive leading coefficient satisfies the following conditions. (가) The equation $f ( x ) - x = 0$ has exactly 2 distinct real roots. (나) The equation $f ( x ) + x = 0$ has exactly 2 distinct real roots. When $f ( 0 ) = 0$ and $f ^ { \prime } ( 1 ) = 1$, find the value of $f ( 3 )$. [4 points]

For two constants $a$ and $b$ with $a < b$, define the function $f ( x )$ as $$f ( x ) = ( x - a ) ( x - b ) ^ { 2 }$$ For the inverse function $g ^ { - 1 } ( x )$ of the function $g ( x ) = x ^ { 3 } + x + 1$, the composite function $h ( x ) = \left( f \circ g ^ { - 1 } \right) ( x )$ satisfies the following conditions. Find the value of $f ( 8 )$. [4 points] (가) The function $( x - 1 ) | h ( x ) |$ is differentiable on the set of all real numbers. (나) $h ^ { \prime } ( 3 ) = 2$

For a cubic function $f ( x )$ with leading coefficient $\frac { 1 } { 2 }$ and a real number $t$, let $g ( t )$ be the number of real roots of the equation $f ^ { \prime } ( x ) = 0$ in the closed interval $[ t , t + 2 ]$. The function $g ( t )$ satisfies the following conditions.
(a) For all real numbers $a$, $\lim _ { t \rightarrow a + } g ( t ) + \lim _ { t \rightarrow a - } g ( t ) \leq 2$.
(b) $g ( f ( 1 ) ) = g ( f ( 4 ) ) = 2 , g ( f ( 0 ) ) = 1$ Find the value of $f ( 5 )$. [4 points]

The function $f ( x ) = 2 x ^ { 3 } - 9 x ^ { 2 } + a x + 5$ has a local maximum at $x = 1$ and a local minimum at $x = b$. What is the value of $a + b$? (Here, $a$ and $b$ are constants.) [3 points]
(1) 12
(2) 14
(3) 16
(4) 18
(5) 20

A cubic function $f(x)$ with leading coefficient 1 satisfies the following condition.
For the function $f(x)$, $$f(k-1)f(k+1) < 0$$ has no integer solutions for $k$.
If $f'\left(-\frac{1}{4}\right) = -\frac{1}{4}$ and $f'\left(\frac{1}{4}\right) < 0$, find the value of $f(8)$. [4 points]

For a constant $a$ ($a \neq 3\sqrt{5}$) and a quadratic function $f(x)$ with negative leading coefficient, the function $$g(x) = \begin{cases} x^{3} + ax^{2} + 15x + 7 & (x \leq 0) \\ f(x) & (x > 0) \end{cases}$$ satisfies the following conditions. (가) The function $g(x)$ is differentiable on the set of all real numbers. (나) The equation $g'(x) \times g'(x - 4) = 0$ has exactly 4 distinct real roots. What is the value of $g(-2) + g(2)$? [4 points]
(1) 30
(2) 32
(3) 34
(4) 36
(5) 38

For a positive number $a$, let the function $f(x)$ be $$f(x) = 2x^{3} - 3ax^{2} - 12a^{2}x$$ When the local maximum value of $f(x)$ is $\frac{7}{27}$, what is the value of $f(3)$? [3 points]

Given the function $\mathrm { f } ( \mathrm { x } ) = \mathrm { a } x ^ { 3 } + x ^ { 2 } ( \mathrm { a } \in \mathrm { R } )$ has an extremum at $\mathrm { x } = - \frac { 4 } { 3 }$ .
(I) Determine the value of $a$;
(II) Let $\mathrm { g } ( \mathrm { x } ) = \mathrm { f } ( \mathrm { x } ) e ^ { x }$. Discuss the monotonicity of $\mathrm { g } ( \mathrm { x } )$.

Given that $x = x_1$ and $x = x_2$ are the local minimum and local maximum points respectively of the function $f(x) = 2a^x - ex^2$ ($a > 0$ and $a \neq 1$). If $x_1 < x_2$, then the range of $a$ is $\_\_\_\_$.

If $x = 2$ is an extremum point of the function $f(x) = (x-1)(x-2)(x-a)$, then $f(0) = $ \_\_\_\_

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