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 $k$?

4. Let $f$ be the function given by $f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + p$, where $p$ is an arbitrary constant.
(a) Write an expression for $f ^ { \prime } ( x )$ and use it to find the relative maximum and minimum values of $f$ in terms of $p$. Show the analysis that leads to your conclusion.
(b) For what values of the constant $p$ does $f$ have 3 distinct real roots?
(c) Find the value of $p$ such that the average value of $f$ over the closed interval $[ - 1,2 ]$ is 1 . [Figure]

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 $\mathrm { g } ( \mathrm { x } )$.

21. Solution: (1) When $a = 0$, it clearly does not satisfy the problem conditions, so $a \neq 0$. & 1 mark \hline $f'(x) = 3ax^2 - 2x$. Setting $f'(x) = 0$, we get $x = 0$ or $x = \frac{2}{3a}$. & $f'(x) = 3ax^2 - 2x$. Setting $f'(x) = 0$, we get $x = 0$ or $x = \frac{2}{3a}$. \hline From the problem conditions, $1 < \frac{2}{3a} < 3$. Solving, we get $\frac{2}{9} < a < \frac{2}{3}$, i.e., the range of $a$ is $\left(\frac{2}{9}, \frac{2}{3}\right)$. & 4 marks \hline (2) When $a = 0$, $f(x) = -x^2$ has minimum value $f(2) = -4$ on $[-1, 2]$. & 5 marks \hline Since $x \in [-1, 2]$, $\therefore 3x - \frac{2}{a} \leq 0$. & When $0 < a \leq \frac{1}{3}$, $\frac{2}{a} \geq 6$. $f'(x) = ax\left(3x - \frac{2}{a}\right)$. \hline 6 marks & \hline $\because f(2) - f(-1) = (8a - 4) - (-a - 1) = 9a - 3 \leq 0$, $\therefore f(x)_{\min} = f(2) = 8a - 4$. & 7 marks \hline When $a > \frac{1}{3}$, $f'(x) = ax\left(3x - \frac{2}{a}\right)$, $0 < \frac{2}{3a} < 2$. When $x \in [-1, 0) \cup \left(\frac{2}{3a}, 2\right]$, $f'(x) > 0$; & \hline When $x \in \left(0, \frac{2}{3a}\right)$, $f'(x) < 0$. & 8 marks \hline $\therefore f(x)_{\min} = \min\left\{f(-1), f\left(\frac{2}{3a}\right)\right\}$. & \hline 9 marks & \hline Since $a > \frac{1}{3}$, $\therefore 27a^3 + 27a^2 - 4 > 0$, $\frac{27a^3 + 27a^2 - 4}{27a^2} > 0$. & 10 marks \hline $\therefore f(x)_{\min} = f(-1) = -a - 1$. & 11 marks \hline In summary, when $0 \leq a \leq \frac{1}{3}$, $f(x)_{\min} = 8a - 4$; when $a > \frac{1}{3}$, $f(x)_{\min} = -a - 1$. & 12 marks \hline 1 mark & \hline

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) = $ \_\_\_\_

For each value of $a$, the graph of $f _ { a }$ has exactly two extrema. Determine the value of $a$ for which the graph of the function $f _ { a }$ has an extremum at the point $x = 3$.
Given is the function $f : x \mapsto 2 \cdot \left( ( \ln x ) ^ { 2 } - 1 \right)$ defined on $\mathbb { R } ^ { + }$. Figure 1 shows the graph $G _ { f }$ of $f$.
[Figure]
Fig. 1
(1a) [5 marks] Show that $x = e ^ { - 1 }$ and $x = e$ are the only zeros of $f$, and calculate the coordinates of the minimum point $T$ of $G _ { f }$. (for verification: $f ^ { \prime } ( x ) = \frac { 4 } { x } \cdot \ln x$ )
(1b) [6 marks] Show that $G _ { f }$ has exactly one inflection point $W$, and determine its coordinates and the equation of the tangent to $G _ { f }$ at the point $W$. (for verification: $x$-coordinate of $W : e$ )
(1c) [6 marks] Justify that $\lim _ { x \rightarrow 0 } f ^ { \prime } ( x ) = - \infty$ and $\lim _ { x \rightarrow + \infty } f ^ { \prime } ( x ) = 0$ hold. Give $f ^ { \prime } ( 0,5 )$ and $f ^ { \prime } ( 10 )$ to one decimal place and draw the graph of the derivative function $f ^ { \prime }$ taking into account all previous results in Figure 1.
(1d) [3 marks] Justify using Figure 1 that there are two values $c \in ] 0 ; 6 ]$ for which $\int _ { e ^ { - 1 } } ^ { c } f ( x ) \mathrm { dx } = 0$ holds.
The rational function $h : x \mapsto 1,5 x - 4,5 + \frac { 1 } { x }$ with $x \in \mathbb { R } \backslash \{ 0 \}$ provides a good approximation for $f$ in a certain range.
(1e) [2 marks] Specify the equations of the two asymptotes of the graph of $h$.
(1f) [5 marks] In the fourth quadrant, $G _ { f }$ together with the $x$-axis and the lines with equations $x = 1$ and $x = 2$ enclose a region whose area is approximately 1.623. Determine the percentage deviation from this value if the function $h$ is used as an approximation for the function $f$ when calculating the area.
By reflecting $G _ { f }$ across the line $x = 4$, the graph of a function $g$ defined on $] - \infty ; 8 [$ is created. This graph is denoted by $G _ { g }$. (2a) [2 marks] Draw $G _ { g }$ in Figure 1.
(2b) [3 marks] The described reflection of $G _ { f }$ across the line $x = 4$ can be replaced by a reflection of $G _ { f }$ across the $y$-axis followed by a translation. Describe this translation and specify $a , b \in \mathbb { R }$ such that $g ( x ) = f ( a x + b )$ for $x \in ] - \infty ; 8 [$.
In the following, the ``w-shaped'' curve $k$ is considered, which consists of the part of $G _ { f }$ restricted to $0,2 \leq x \leq 4$ and the part of $G _ { g }$ restricted to $4 < x \leq 7,8$. The curve $k$ is translated 12 units in the negative $z$-direction. The area swept out in this process serves as a model for a 12-meter-long aquarium, which is completed by two flat walls at the front and back to form a basin (see Figure 2). Here, one unit of length in the coordinate system corresponds to one meter in reality.
[Figure]
Fig. 2
(2c) [3 marks] The aquarium walls form a tunnel at the bottom through which visitors can walk. Calculate the size of the angle that the left and right tunnel walls enclose with each other.
The aquarium is completely filled with water.
(2d) [2 marks] Calculate the maximum water depth of the aquarium.
\footnotetext{(c) \href{http://Abiturloesung.de}{Abiturloesung.de} }
(2e) [3 marks] The volume of water in the aquarium can be calculated analogously to the volume of a prism with base area $G$ and height $h$. Explain that the term $24 \cdot \int _ { 0,2 } ^ { 4 } ( f ( 0,2 ) - f ( x ) ) \mathrm { dx }$ describes the water volume in the completely filled aquarium in cubic meters.