Consider the region $R = \left\{(x, y) : x \leq y \leq 9 - \frac{11}{3}x^2,\, x \geq 0\right\}$. The area of the largest rectangle of sides parallel to the coordinate axes and inscribed in $R$, is:
(1) $\frac{730}{119}$
(2) $\frac{625}{111}$
(3) $\frac{821}{123}$
(4) $\frac{567}{121}$

Suppose that $x$ and $y$ satisfy
$$3 x + y = 18 , \quad x \geqq 1 , \quad y \geqq 6 .$$
We are to find the maximum value and the minimum value of $x y$.
When we express $x y$ in terms of $x$, we have
$$x y = \mathbf { A B } ( x - \mathbf { C } ) ^ { 2 } + \mathbf { D E } .$$
Also, the range of values which $x$ can take is
$$\mathbf { F } \leqq x \leqq \mathbf { G } .$$
Hence, the value of $x y$ is maximized at $x = \mathbf { H }$ and its value there is $\mathbf { I J }$, and the value of $x y$ is minimized at $x = \mathbf { K }$ and its value there is $\mathbf { L M }$.

Suppose that $x$ and $y$ satisfy
$$3 x + y = 18 , \quad x \geqq 1 , \quad y \geqq 6$$
We are to find the maximum value and the minimum value of $x y$.
When we express $x y$ in terms of $x$, we have
$$x y = \mathbf { A B } ( x - \mathbf { C } ) ^ { 2 } + \mathbf { D E } .$$
Also, the range of values which $x$ can take is
$$\mathbf { F } \leq x \leqq \mathbf { G }$$
Hence, the value of $x y$ is maximized at $x = \mathbf { H }$ and its value there is $\mathbf { I J }$, and the value of $x y$ is minimized at $x = \mathbf { K }$ and its value there is $\mathbf { L M }$.

Let $a$ be a positive real number. We are to investigate local extrema of the function
$$f(x) = x^2 - 5 + 4a\log(2x + a + 8) \quad \left(-\frac{a}{2} - 4 < x < -2\right).$$
(1) When we differentiate the function $f(x)$ with respect to $x$, we obtain
$$f'(x) = \frac{\mathbf{A}(\mathbf{B}\, x + a)(x + \mathbf{C})}{\mathbf{D}\, x + a + \mathbf{E}}.$$
(2) Since a condition of $a$ is that $a > 0$ and the domain of $f(x)$ is $-\frac{a}{2} - 4 < x < -2$, the range of values of $a$ such that $f(x)$ has both a local maximum and a local minimum is
$$\mathbf{F} < a < \mathbf{G}.$$
In such a case, the sum of the local maximum and the local minimum is
$$\frac{a^2}{\mathbf{H}} + \mathbf{I} + \mathbf{IJ}\, a\log\mathbf{K}\, a.$$

We are to find the range of the values of a real number $t$ such that the maximum value of the cubic function $$f(x) = \frac{1}{3}x^3 - \frac{t+2}{2}x^2 + 2tx + \frac{2}{3}$$ over the interval $x \leqq 4$ is greater than 6.
First of all, since the derivative of $f(x)$ is $$f'(x) = (x - \mathbf{A})(x - t),$$ we consider the problem by dividing the range of the values of $t$ as follows:
(i) When $t > \mathbf{A}$, $f(x)$ has a local maximum at $x = \mathbf{A}$ and a local minimum at $x = t$. Since $f(4) = \mathbf{B}$, we only have to find the range of the values of $t$ satisfying $f(\mathbf{A}) > 6$.
(ii) When $t = \mathbf{A}$, the maximum value of $f(x)$ over the interval $x \leqq 4$ is $f(\mathbf{C}) = \mathbf{D}$, and hence the condition is not satisfied.
(iii) When $t < \mathbf{A}$, $f(x)$ has a local maximum at $x = t$ and a local minimum at $x = \mathbf{A}$. Since $f(4) = \mathbf{B}$, we only have to find the range of the values of $t$ satisfying $f(t) > 6$.
Here, we note $$f(t) - 6 = -\frac{1}{6}(t + \mathbf{E})(t - \mathbf{EF})^2.$$
From the above, the range of the values of $t$ is to be determined.

Consider all segments PQ of length 2 such that the end points P and Q are on the parabola $y = x ^ { 2 }$. Denote the mid-point of the segment PQ by M. Among all M, we are to find the coordinates of the ones nearest to the $x$-axis.
Let us denote the coordinates of the end points of segment PQ by $\mathrm{ P }\left( p , p ^ { 2 } \right)$ and $\mathrm{ Q }\left( q , q ^ { 2 } \right)$. Then the $y$-coordinate $m$ of M is
$$m = \frac { p ^ { 2 } + q ^ { 2 } } { \mathbf { M } } .$$
Next, since $\mathrm{ PQ } = 2$, then
$$( p - q ) ^ { 2 } + \left( p ^ { 2 } - q ^ { 2 } \right) ^ { 2 } = \mathbf { N }$$
by the Pythagorean theorem.
Now, when we set $t = p q$, we obtain from (1) and (2) the quadratic equation in $m$
$$\mathbf { O } m ^ { 2 } + m - \mathbf { P } t ^ { 2 } - t - \mathbf { Q } = 0 .$$
When we solve this for $m$, noting that $m > 0$, we have
$$m = - \frac { 1 } { \mathbf { R } } + \sqrt { \left( t + \frac { 1 } { \mathbf { S } } \right) ^ { 2 } + \mathbf{T} } .$$
This shows that $m$ is minimized when $t = - \dfrac { 1 } { \mathbf{U} }$. In this case, $p q = - \dfrac { 1 } { \mathbf{U} }$ and $p ^ { 2 } + q ^ { 2 } = \dfrac { \mathbf { V } } { \mathbf { V } }$, and so we have $p + q = \pm \mathbf { W }$.
Thus the coordinates of the M nearest to the $x$-axis are $\left( \pm \dfrac { 1 } { \mathbf { X } } , \dfrac { \mathbf { Y } } { \mathbf { Z } } \right)$.

Consider a rhombus ABCD with sides of length $a$, where $a$ is a constant. Let $r$ be the radius of the circle O inscribed in the rhombus ABCD, and $\mathrm{K}, \mathrm{L}$, $\mathrm{M}, \mathrm{N}$ be the points of tangency of the circle O and the rhombus. Let $S$ denote the area of the part of the rhombus outside circle O.
We are to find the range of the values of $r$, and the maximum value of $S$.
(1) For each of $\mathbf{A}$ $\sim$ $\mathbf{C}$ below, choose the correct answer from among (0) $\sim$ (9).
Let $\angle \mathrm{ABO} = \theta$. We have $\mathrm{OB} = \mathbf{A}$, and hence $\mathrm{OK} = \mathbf{B}$. Hence, since $( \cos \theta - \sin \theta ) ^ { 2 } \geqq 0$, the range of the values taken by $r$ is
$$0 < r \leqq \mathbf { C } .$$
(0) $a$ (1) $\frac { a } { 2 }$ (2) $\frac { a } { 3 }$ (3) $a \sin \theta$ (4) $a \cos \theta$ (5) $a \tan \theta$ (6) $a \sin ^ { 2 } \theta$ (7) $a \cos ^ { 2 } \theta$ (8) $a \sin \theta \cos \theta$ (9) $a \tan ^ { 2 } \theta$
(2) For each of $\mathbf { D } \sim$ $\mathbf{F}$ below, choose the correct answer from among (0) $\sim$ (9).
When the area $S$ is expressed in terms of $r$, we have
$$S = \mathbf { D } .$$
Here, we observe that when $r = \mathbf { E }$, the value of $\mathbf{D}$ is maximized, and this value for $r$ satisfies (1). Thus, at $r = \mathbf { E }$, $S$ takes the maximum value $\mathbf { F }$.
(0) $2 a r - \pi r ^ { 2 }$ (1) $a r - \frac { \pi } { 2 } r ^ { 2 }$ (2) $\frac { a } { 2 } r - \pi r ^ { 2 }$ (3) $\frac { a r - \pi r ^ { 2 } } { 2 }$ (4) $\frac { 2 a } { \pi }$ (5) $\frac { a } { \pi }$ (6) $\frac { a } { 2 \pi }$ (7) $\frac { 4 a ^ { 2 } } { \pi }$ (8) $\frac { a ^ { 2 } } { \pi }$ (9) $\frac { a ^ { 2 } } { 4 \pi }$

Q2 Consider the quadratic function
$$f ( x ) = \frac { 3 } { 4 } x ^ { 2 } - 3 x + 4$$
Let $a$ and $b$ be real numbers satisfying $0 < a < b$ and $2 < b$. We are to find the values of $a$ and $b$ such that the range of the values of the function $y = f ( x )$ on $a \leqq x \leqq b$ is $a \leqq y \leqq b$.
Since the equation of the axis of symmetry of the graph of $y = f ( x )$ is $x = \mathbf { M }$, we divide the problem into two cases as follows:
(i) $\mathbf{M} \leqq a$;
(ii) $0 < a < \mathbf{M}$.
In the case of (i), since the values of $f ( x )$ increase with $x$ on $a \leqq x \leqq b$, the equations $f ( a ) = a$ and $f ( b ) = b$ have to be satisfied. By solving these, we obtain $a = \frac { \mathbf { N } } { \mathbf { O } }$ and $b = \mathbf { P }$. However, this $a$ does not satisfy (i).
In the case of (ii), since the minimum value of $f ( x )$ on $a \leqq x \leqq b$ is $\mathbf { Q }$, we have
$$a = \mathbf { R } .$$
This satisfies (ii). Then since $f ( a ) = \frac { \mathbf { S } } { \mathbf { T } } < b$, we have $f ( b ) = b$. Hence, we obtain
$$b = \mathbf { U } .$$

Given the function
$$f ( x ) = x ^ { 3 } - 3 a x ^ { 2 } - 3 ( 2 a + 1 ) x + a + 2 ,$$
answer the following questions.
(1) For $\mathbf { G }$ $\sim$ $\mathbf { K }$, choose the correct answers from among (0) $\sim$ (5) below, and for the other $\square$, enter the correct numbers.
Since
$$f ^ { \prime } ( x ) = \mathbf { A } ( x - \mathbf { B } a - \mathbf { C } ) ( x + \mathbf { D } ) ,$$
we see that
(i) when $a > \mathbf { EF }$, $f ( x )$ is $\mathbf { G }$ at $x = - \square \mathbf { D }$ and is $\square$ H at $x =$ $\square$ B $a +$ $\square$ C;
(ii) when $a =$ $\square$ EF, $f ( x )$ is always $\square$ I;
(iii) when $a < \mathbf{EF}$, $f ( x )$ is $\square$ J at $x = -$ $\square$ D and is $\square$ K at $x =$ $\square$ B $a +$ $\square$ C. (0) locally maximized
(1) locally minimized
(2) increasing
(3) decreasing
(4) maximized
(5) minimized
(2) When we express the minimum value $m$ of $f ( x )$ over the range $- 1 \leqq x \leqq 1$ in terms of $a$, we have that
(i) when $a \geqq \mathbf { L }$, $m = \mathbf { MN } a$;
(ii) when $\mathbf { OP } \leqq a < \mathbf { L }$, $m = \mathbf { QR } \left( a ^ { 3 } + \mathbf { S } a ^ { 2 } + \mathbf { T } a \right)$;
(iii) when $a < \mathbf{OP}$, $m = \mathbf { U } a + \mathbf { V }$.
(3) The value of $m$ in (2) is maximized at $a = \frac { - \mathbf { W } + \sqrt { \mathbf { X } } } { \square \mathbf { Y } }$.

Let $a$ be a real number satisfying $a \geqq 0$. We are to express the maximum value $M$ of the function $f(x) = |x^2 - 2x|$ on the range $a \leqq x \leqq a + 1$ in terms of $a$. Furthermore, we are to find the minimum value of $M$ over the range $a \geqq 0$.
(1) The function $f(x)$ can be expressed without using the absolute value symbol as follows: when $x \leqq \mathbf{M}$ or $x \geqq \mathbf{N}$, then $f(x) = x^2 - 2x$; when $\mathbf{M} < x < \mathbf{N}$, then $f(x) = -x^2 + 2x$.
Hence, the maximum value of $f(x)$ on $a \leqq x \leqq a + 1$ is the following: when $0 \leqq a \leqq \mathbf{O}$, then $M = \mathbf{P}$; when $\mathbf{O} < a \leqq \frac{\mathbf{Q} + \sqrt{\mathbf{R}}}{\mathbf{S}}$, then $M = -a^2 + \frac{\mathbf{T}}{\mathbf{S}}a$; when $a > \frac{\mathbf{Q} + \sqrt{\mathbf{R}}}{\mathbf{S}}$, then $M = a^2 - \mathbf{U}$.
(2) The minimum value of $M$ over the range $a \geqq 0$ is $\frac{\sqrt{\mathbf{V}}}{\mathbf{W}}$.

Consider vectors $\vec{a}$ and $\vec{b}$ on the coordinate plane satisfying $|\vec{a}| + |\vec{b}| = 9$ and $|\vec{a} - \vec{b}| = 7$. Let $|\vec{a}| = x$, where $1 < x < 8$, and let the angle between $\vec{a}$ and $\vec{b}$ be $\theta$. Using the triangle formed by vectors $\vec{a}$, $\vec{b}$, and $\vec{a} - \vec{b}$, we can express $\cos\theta$ in terms of $x$ as $f(x) = \frac{c}{9x - x^2} + d$, where $c$ and $d$ are constants with $c > 0$, with domain $\{x \mid 1 < x < 8\}$. Explain where $f(x)$ is increasing and decreasing in its domain. Determine the value of $x$ for which the angle $\theta$ between $\vec{a}$ and $\vec{b}$ is maximum. (Non-multiple choice question, 4 points)

Consider the real coefficient polynomial $f ( x ) = x ^ { 4 } - 4 x ^ { 3 } - 2 x ^ { 2 } + a x + b$. It is known that the equation $f ( x ) = 0$ has a complex root $1 + 2 i$ (where $i = \sqrt { - 1 }$). Select the correct options.
(1) $1 - 2i$ is also a root of $f ( x ) = 0$
(2) Both $a$ and $b$ are positive numbers
(3) $f ^ { \prime } ( 2.1 ) < 0$
(4) The function $y = f ( x )$ has a local minimum at $x = 1$
(5) The $x$-coordinates of all inflection points of the graph $y = f ( x )$ are greater than 0

A real coefficient polynomial $f(x)$ has degree greater than 5, and its leading coefficient is positive. Moreover, $f(x)$ has local minima at $x = 1, 2, 4$ and local maxima at $x = 3, 5$. Based on the above, select the correct options.
(1) $f(1) < f(3)$
(2) There exist real numbers $a, b$ satisfying $1 < a < b < 2$ such that $f'(a) > 0$ and $f'(b) < 0$
(3) $f''(3) > 0$
(4) There exists a real number $c > 5$ such that $f'(c) > 0$
(5) The degree of $f(x)$ is greater than 7

Let $f ( x )$ be a real-coefficient cubic polynomial. The function $y = f ( x )$ has a local minimum at $x = - 3$ and a local maximum at $x = 1$.
Regarding the statements about $f ^ { \prime \prime } ( - 3 )$ and $f ^ { \prime \prime } ( 1 )$, select the correct option. (Single choice)
(1) $f ^ { \prime \prime } ( - 3 ) = f ^ { \prime \prime } ( 1 ) = 0$
(2) $f ^ { \prime \prime } ( - 3 ) > 0$ and $f ^ { \prime \prime } ( 1 ) > 0$
(3) $f ^ { \prime \prime } ( - 3 ) > 0$ and $f ^ { \prime \prime } ( 1 ) < 0$
(4) $f ^ { \prime \prime } ( - 3 ) < 0$ and $f ^ { \prime \prime } ( 1 ) > 0$
(5) $f ^ { \prime \prime } ( - 3 ) < 0$ and $f ^ { \prime \prime } ( 1 ) < 0$

Consider an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ in the $xy$-plane. Here, $a$ and $b$ are constants satisfying $a > b > 0$. Answer the following questions.

1. Find the equation of the tangent line at a point $(X, Y)$ on the ellipse in the first quadrant.
2. The tangent line obtained in Question I. 1 intersects the $x$- and $y$-axes. Find the coordinates $(X, Y)$ at the tangent point that minimizes the length of the segment connecting the two intersects and obtain the minimum length of the segment.
3. Consider a region bounded by the segment obtained in Question I. 2 and the $x$- and $y$-axes, and let $C_{1}$ be a cone formed by rotating the region around the $x$-axis. Next, let $C_{2}$ be a cone with the maximum volume while having the same surface area (including a base area) as the cone $C_{1}$. Find $\frac{S_{2}}{S_{1}}$, where $S_{1}$ and $S_{2}$ are the base areas of the cones $C_{1}$ and $C_{2}$, respectively.

The function $f(x) = mx - 1 + \frac{1}{x}$ is given.
Accordingly, what is the smallest value of $m$ that satisfies the property $f(x) \geq 0$ for all $x > 0$?
A) $\frac{1}{2}$
B) $\frac{1}{3}$
C) $\frac{1}{4}$
D) $\frac{1}{5}$
E) $\frac{1}{6}$

$$f(x) = x^{4} - 5x^{2} + 4$$
What is the maximum value of the function on the interval $\left[\frac{-1}{2}, \frac{1}{2}\right]$?
A) 8
B) 6
C) 4
D) 2
E) 0

A workplace consisting of a corridor, kitchen, and study room has the model shown above as rectangle ABCD, and the perimeter of this rectangle is 72 meters.
For the kitchen in this workplace to have the largest area, what should $x$ be in meters?
A) 1
B) 2
C) 3
D) 4
E) 5

A line $d$ with negative slope passing through the point $(1,2)$ forms a triangular region with the coordinate axes. What is the minimum area of this triangular region in square units?
A) 2
B) 3
C) 4
D) $\frac { 9 } { 2 }$
E) $\frac { 7 } { 2 }$

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) $\frac { -10 } { 3 }$

For $x > 0$; if the point $(a, b)$ on the graph of the curve $y = 6 - x^2$ is closest to the point $(0, 1)$, what is b?
A) $\frac { 3 } { 2 }$
B) $\frac { 5 } { 2 }$
C) $\frac { 7 } { 2 }$
D) $\frac { 9 } { 2 }$
E) $\frac { 11 } { 2 }$

A tour company charges 140 TL per person for a tour it will organize. If the number of registered participants exceeds 80, a refund of 50 kuruş will be made to all participants for each person above 80. The capacity is limited to 200 people.
For example, if 100 people participate in the tour, everyone receives a 10 TL refund and the per-person fee is 130 TL.
Accordingly, how many people should participate in the tour so that the company's revenue from participants is maximum?
A) 160
B) 165
C) 175
D) 180
E) 185

The graph of the derivative function $f ^ { \prime }$ of a function f defined on the set of real numbers is given below.
Accordingly, regarding the function f: I. It is decreasing. II. $f ( a )$ is a local maximum value. III. $f ^ { \prime \prime } ( a )$ is not defined.
Which of the following statements are true?
A) Only I
B) Only II
C) I and III
D) II and III
E) I, II and III

In the rectangular coordinate plane, the graph of the curve $y = e ^ { \left( - x ^ { 2 } \right) }$ is given.
In this plane, a rectangle with one side on the x-axis and two vertices on the curve is drawn with the maximum possible area.
What is the area of this rectangle in square units?
A) $\sqrt { \mathrm { e } }$
B) $\sqrt { 2 e }$
C) $\frac { \sqrt { e } } { 2 }$
D) $\sqrt { \frac { 2 } { \mathrm { e } } }$
E) $2 \sqrt { e }$

Let $f$ be a function defined on the set of real numbers, and let the derivative of $f$ be denoted by $f ^ { \prime }$. The graph of the function $f ^ { \prime }$ is the parabolic curve shown in the figure.
Accordingly, regarding the function f: I. $f ( 0 ) < 0$ II. It is decreasing on the interval (-a, a). III. $f ( a )$ is a local minimum value.
Which of the following statements are definitely true?
A) Only II
B) Only III
C) I and II
D) II and III
E) I, II and III