For the quadratic function $f ( x ) = \frac { 3 x - x ^ { 2 } } { 2 }$, a function $g ( x )$ defined on the interval $[ 0 , \infty )$ satisfies the following conditions. (가) When $0 \leq x < 1$, $g ( x ) = f ( x )$. (나) When $n \leq x < n + 1$, $$\begin{aligned} & g ( x ) = \frac { 1 } { 2 ^ { n } } \{ f ( x - n ) - ( x - n ) \} + x \\ & \text{(Here, } n \text{ is a natural number.)} \end{aligned}$$ For some natural number $k ( k \geq 6 )$, the function $h ( x )$ is defined as $$h ( x ) = \begin{cases} g ( x ) & ( 0 \leq x < 5 \text{ or } x \geq k ) \\ 2 x - g ( x ) & ( 5 \leq x < k ) \end{cases}$$ When the sequence $\left\{ a _ { n } \right\}$ is defined by $a _ { n } = \int _ { 0 } ^ { n } h ( x ) d x$ and $\lim _ { n \rightarrow \infty } \left( 2 a _ { n } - n ^ { 2 } \right) = \frac { 241 } { 768 }$, find the value of $k$. [4 points]

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

A cubic function $f ( x )$ with leading coefficient 1 and a quadratic function $g ( x )$ with leading coefficient $-1$ satisfy the following conditions. (가) The tangent line to the curve $y = f ( x )$ at the point $( 0,0 )$ and the tangent line to the curve $y = g ( x )$ at the point $( 2,0 )$ are both the $x$-axis. (나) The number of tangent lines to the curve $y = f ( x )$ drawn from the point $( 2,0 )$ is 2. (다) The equation $f ( x ) = g ( x )$ has exactly one real root. For all real numbers $x > 0$, $$g ( x ) \leq k x - 2 \leq f ( x )$$ Let $\alpha$ and $\beta$ be the maximum and minimum values of the real number $k$ satisfying the above inequality, respectively. When $\alpha - \beta = a + b \sqrt { 2 }$, find the value of $a ^ { 2 } + b ^ { 2 }$. (Here, $a$ and $b$ are rational numbers.) [4 points]

How many integers $a$ are there such that the curve $y = a x ^ { 2 } - 2 \sin 2 x$ has an inflection point? [3 points]
(1) 4
(2) 5
(3) 6
(4) 7
(5) 8

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 a real number $a$ ($a > 1$), define the function $f ( x )$ as $$f ( x ) = ( x + 1 ) ( x - 1 ) ( x - a)$$ Define the function $$g ( x ) = x ^ { 2 } \int _ { 0 } ^ { x } f ( t ) d t - \int _ { 0 } ^ { x } t ^ { 2 } f ( t ) d t$$ such that $g ( x )$ has exactly one extremum. What is the maximum value of $a$? [4 points]
(1) $\frac { 9 \sqrt { 2 } } { 8 }$
(2) $\frac { 3 \sqrt { 6 } } { 4 }$
(3) $\frac { 3 \sqrt { 2 } } { 2 }$
(4) $\sqrt { 6 }$
(5) $2 \sqrt { 2 }$

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) $h ^ { \prime } ( 3 ) = 2$

The function $f ( x )$ is a cubic function with leading coefficient 1, and the function $g ( x )$ is a linear function. Define the function $h ( x )$ as $$h ( x ) = \begin{cases} | f ( x ) - g ( x ) | & ( x < 1 ) \\ f ( x ) + g ( x ) & ( x \geq 1 ) \end{cases}$$ If $h ( x )$ is differentiable on the entire set of real numbers, and $h ( 0 ) = 0$, $h ( 2 ) = 5$, find the value of $h ( 4 )$. [4 points]

How many integers $k$ are there such that the equation $2 x ^ { 3 } - 3 x ^ { 2 } - 12 x + k = 0$ has three distinct real roots? [3 points]
(1) 20
(2) 23
(3) 26
(4) 29
(5) 32

Find the maximum value of the real number $a$ such that the function $f ( x ) = x ^ { 3 } + a x ^ { 2 } - \left( a ^ { 2 } - 8 a \right) x + 3$ is increasing on the entire set of real numbers. [3 points]

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

Find the number of integers $k$ such that the equation $2 x ^ { 3 } - 6 x ^ { 2 } + k = 0$ has exactly 2 distinct positive real roots. [3 points]

A cubic function $f ( x )$ with leading coefficient 1 and a function $g ( x )$ that is continuous on the set of all real numbers satisfy the following conditions. Find the value of $f ( 4 )$. [4 points] (가) For all real numbers $x$, $$f ( x ) = f ( 1 ) + ( x - 1 ) f ^ { \prime } ( g ( x ) )$$ (나) The minimum value of the function $g ( x )$ is $\frac { 5 } { 2 }$. (다) $f ( 0 ) = - 3$, $f ( g ( 1 ) ) = 6$

For the function $f(x) = \frac{1}{3}x^3 - 2x^2 - 12x + 4$, if $f$ has a local maximum at $x = \alpha$ and a local minimum at $x = \beta$, find the value of $\beta - \alpha$. (Here, $\alpha$ and $\beta$ are constants.) [3 points]
(1) $-4$
(2) $-1$
(3) 2
(4) 5
(5) 8

For a positive number $a$, the function $f(x)$ defined on $x \geq -1$ is $$f(x) = \begin{cases} -x^2 + 6x & (-1 \leq x < 6) \\ a\log_4(x-5) & (x \geq 6) \end{cases}$$ For a real number $t \geq 0$, let $g(t)$ denote the maximum value of $f(x)$ on the closed interval $[t-1, t+1]$. If the minimum value of the function $g(t)$ on the interval $[0, \infty)$ is 5, find the minimum value of the positive number $a$. [4 points]

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]

The function $f ( x )$ is $$f ( x ) = \begin{cases} - x ^ { 2 } & ( x < 0 ) \\ x ^ { 2 } - x & ( x \geq 0 ) \end{cases}$$ and for a positive number $a$, the function $g ( x )$ is $$g ( x ) = \left\{ \begin{array} { c l } a x + a & ( x < - 1 ) \\ 0 & ( - 1 \leq x < 1 ) \\ a x - a & ( x \geq 1 ) \end{array} \right.$$ Let $k$ be the maximum value of $a$ such that the function $h ( x ) = \int _ { 0 } ^ { x } ( g ( t ) - f ( t ) ) d t$ has exactly one extremum. When $a = k$, what is the value of $k + h ( 3 )$? [4 points]
(1) $\frac { 9 } { 2 }$
(2) $\frac { 11 } { 2 }$
(3) $\frac { 13 } { 2 }$
(4) $\frac { 15 } { 2 }$
(5) $\frac { 17 } { 2 }$

For all real numbers $x$ with $- 2 \leq x \leq 2$, the inequality $$- k \leq 2 x ^ { 3 } + 3 x ^ { 2 } - 12 x - 8 \leq k$$ holds. Find the minimum value of the positive number $k$. [3 points]

9. If the function $f ( x ) = \frac { 1 } { 2 } ( m - 2 ) x ^ { 2 } + ( n - 8 ) x + 1$ $(m \geq 0, n \geq 0)$ is monotonically decreasing on the interval $\left[ \frac { 1 } { 2 }, 2 \right]$, then the maximum value of $m n$ is
(A) $16$
(B) $18$
(C) $25$
(D) $\frac { 81 } { 2 }$

Let $\mathrm { f } ^ { \prime } ( \mathrm { x } )$ be the derivative of the odd function $f ( x ) ( x \in \mathbf { R } )$. Given $\mathrm { f } ( - 1 ) = 0$, and when $\mathrm { x } > 0$, $x f ^ { \prime } ( x ) - f ( x ) < 0$. Then the range of $x$ for which $f ( x ) > 0$ holds is
(A) $( - \infty , - 1 ) \cup ( 0,1 )$
(B) $( - 1,0 ) \cup ( 1 , + \infty )$
(C) $( - \infty , - 1 ) \cup ( - 1,0 )$
(D) $( 0,1 ) \cup ( 1 , + \infty )$

17. Let a be a real number. The maximum value of the function $f ( x ) = \left| x ^ { 2 } - a x \right|$ on the interval $[ 0,1 ]$ is denoted by $g ( a )$. When $a =$ $\_\_\_\_$,
$$\mathbf { y }$$
the value of $g ( a )$ is minimized. III. Solution Questions