Let $g$ be a function such that all its derivatives exist. We say $g$ has an inflection point at $x_0$ if the second derivative $g''$ changes sign at $x_0$ i.e., if $g''(x_0 - \epsilon) \times g''(x_0 + \epsilon) < 0$ for all small enough positive $\epsilon$.
(a) If $g''(x_0) = 0$ then $g$ has an inflection point at $x_0$. True or False?
(b) If $g$ has an inflection point at $x_0$ then $g''(x_0) = 0$. True or False?
(c) Find all values $x_0$ at which $x^{4}(x - 10)$ has an inflection point.

You have a piece of land close to a river, running straight. You are required to cut off a rectangular portion of the land, with the river forming one of the sides of the rectangle so, your fence will have three sides to it. You only have 60 meters of fencing. The maximum area that you can enclose is \_\_\_\_

(a) Compute $\dfrac{d}{dx}\left[\int_{0}^{e^{x}} \log(t)\cos^{4}(t)\,dt\right]$.
(b) For $x > 0$ define $F(x) = \int_{1}^{x} t\log(t)\,dt$.
i. Determine the open interval(s) (if any) where $F(x)$ is decreasing and the open interval(s) (if any) where $F(x)$ is increasing.
ii. Determine all the local minima of $F(x)$ (if any) and the local maxima of $F(x)$ (if any).

A stationary point of a function $f$ is a real number $r$ such that $f ^ { \prime } ( r ) = 0$. A polynomial need not have a stationary point (e.g. $x ^ { 3 } + x$ has none). Consider a polynomial $p ( x )$.
(a) If $p ( x )$ is of degree 2022, then $p ( x )$ must have at least one stationary point.
(b) If the number of distinct real roots of $p ( x )$ is 2021, then $p ( x )$ must have at least 2020 stationary points.
(c) If the number of distinct real roots of $p ( x )$ is 2021, then $p ( x )$ can have at most 2020 stationary points.
(d) If $r$ is a stationary point of $p ( x )$ AND $p ^ { \prime \prime } ( r ) = 0$, then the point $( r , p ( r ) )$ is neither a local maximum nor a local minimum point on the graph of $p ( x )$.

[14 points] For a positive integer $n$, let $f(x) = \sum_{i=0}^{n} x^i = 1 + x + x^2 + \cdots + x^n$. Find the number of local maxima of $f(x)$. Find the number of local minima of $f(x)$. For each maximum/minimum $(c, f(c))$, find the integer $k$ such that $k \leq c < k+1$.
Hints (use these or your own method): It may be helpful to (i) look at small $n$, (ii) use the definition of $f$ as well as a closed formula, and (iii) treat $x \geq 0$ and $x < 0$ separately.

Let $a$ be a point in the domain of a continuous real valued function $f$. One says that $f$ has a flex point at $a$ if we can find a small interval $(a - \epsilon , a + \epsilon)$ in the domain of $f$ such that the following happens: (i) for all $x$ in the open interval $(a - \epsilon , a)$ the sign of $f ^ { \prime \prime } ( x )$ is constant and, (ii) for all $x$ in the open interval $(a , a + \epsilon)$ the sign of $f ^ { \prime \prime } ( x )$ is constant and opposite to the sign of $f ^ { \prime \prime } ( x )$ in $(a - \epsilon , a)$.
Statements
(29) If $f$ is a cubic polynomial with a local maximum at $x = p$ and a local minimum at $x = q$, then $f$ has a unique flex point at $x = \frac { p + q } { 2 }$. (30) If $f ^ { \prime \prime } ( a ) = 0$ then $f$ must have a flex point at $a$. (31) Let $f ( x ) = x ^ { 2 }$ for $x \geq 0$ and $f ( x ) = - x ^ { 2 }$ for $x < 0$. Then $f$ has no flex points. (32) If $f ^ { \prime }$ is monotonic on an open interval $I$, then $f$ cannot have a flex point in $I$.

For $a > 1$, consider the function $f ( x ) = 2 x ^ { 3 } - 3 ( a + 1 ) x ^ { 2 } + 6 a x - 4 a + 2$. Let $b$ be one real root of the equation $f ( x ) = 0$. The following is a process for comparing the magnitudes of two numbers $a$ and $b$. $f ^ { \prime } ( x ) =$ (가) and since $a > 1$, $f ( x )$ has a (나) at $x = 1$. Since $f ( 1 ) < 0$ and $f ( b ) = 0$, $a$ (다) $b$.
What are the correct expressions for (가), (나), and (다) in the above process? [3 points]

(가)	(나)	(다)
(1) $6 ( x - 1 ) ( x - a )$	local minimum	$>$
(2) $6 ( x - 1 ) ( x - a )$	local minimum	$<$
(3) $6 ( x - 1 ) ( x - a )$	local maximum	$>$
(4) $6 ( x - a ) ( x - 1 )$	local maximum	$<$
(5) $6 ( x - a ) ( x - 1 )$	local maximum	$>$

The cubic equation in $x$, $\frac { 1 } { 3 } x ^ { 3 } - x = k$, has three distinct real roots $\alpha$, $\beta$, $\gamma$. For a real number $k$, let $m$ be the minimum value of $| \alpha | + | \beta | + | \gamma |$. Find the value of $m ^ { 2 }$. [4 points]

The function $y = f ( x )$ is continuous on all real numbers, and for all $x$ with $| x | \neq 1$, the derivative $f ^ { \prime } ( x )$ is
$$f ^ { \prime } ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } & ( | x | < 1 ) \\ - 1 & ( | x | > 1 ) \end{array} \right.$$
Which of the following statements in are true? [3 points]

ㄱ. The function $y = f ( x )$ has an extremum at $x = - 1$. ㄴ. For all real numbers $x$, $f ( x ) = f ( - x )$. ㄷ. If $f ( 0 ) = 0$, then $f ( 1 ) > 0$.
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For a natural number $n$, when two points $\mathrm { P } _ { n - 1 } , \mathrm { P } _ { n }$ are on the graph of the function $y = x ^ { 2 }$, the point $\mathrm { P } _ { n + 1 }$ is determined according to the following rule.
(a) The coordinates of the two points $\mathrm { P } _ { 0 } , \mathrm { P } _ { 1 }$ are $(0,0)$ and $(1,1)$, respectively.
(b) The point $\mathrm { P } _ { n + 1 }$ is the intersection of the line passing through point $\mathrm { P } _ { n }$ and perpendicular to the line $\mathrm { P } _ { n - 1 } \mathrm { P } _ { n }$ and the graph of the function $y = x ^ { 2 }$. (Here, $\mathrm { P } _ { n }$ and $\mathrm { P } _ { n + 1 }$ are distinct points.) Let $l _ { n } = \overline { \mathrm { P } _ { n - 1 } \mathrm { P } _ { n } }$. What is the value of $\lim _ { n \rightarrow \infty } \frac { l _ { n } } { n }$? [3 points]
(1) $2 \sqrt { 3 }$
(2) $2 \sqrt { 2 }$
(3) 2
(4) $\sqrt { 3 }$
(5) $\sqrt { 2 }$

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]

There is a quartic function $f ( x )$ with leading coefficient 1, $f ( 0 ) = 3$, and $f ^ { \prime } ( 3 ) < 0$. For a real number $t$, define the set $S$ as $$S = \{ a \mid \text{the function } | f ( x ) - t | \text{ is not differentiable at } x = a \}$$ and let $g ( t )$ be the number of elements in set $S$. When the function $g ( t )$ is discontinuous only at $t = 3$ and $t = 19$, find the value of $f ( - 2 )$. [4 points]

As shown in the figure, there are two points $\mathrm { A } ( - 1,0 )$ and $\mathrm { P } ( t , t + 1 )$ on the line $y = x + 1$. Let Q be the point where the line passing through P and perpendicular to the line $y = x + 1$ meets the $y$-axis. What is the value of $\lim _ { t \rightarrow \infty } \frac { \overline { \mathrm { AQ } } ^ { 2 } } { \overline { \mathrm { AP } } ^ { 2 } }$? [3 points]
(1) 1
(2) $\frac { 3 } { 2 }$
(3) 2
(4) $\frac { 5 } { 2 }$
(5) 3

As shown in the figure, let Q be the foot of the perpendicular from point P on a circle with center O and diameter AB of length 2 to the line segment AB, let R be the foot of the perpendicular from point Q to the line segment OP, and let S be the foot of the perpendicular from point O to the line segment AP. When $\angle \mathrm { PAQ } = \theta \left( 0 < \theta < \frac { \pi } { 4 } \right)$, let $f ( \theta )$ be the area of triangle AOS and $g ( \theta )$ be the area of triangle PRQ. When $\lim _ { \theta \rightarrow +0 } \frac { \theta ^ { 2 } f ( \theta ) } { g ( \theta ) } = \frac { q } { p }$, find the value of $p ^ { 2 } + q ^ { 2 }$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]

A cubic function $f ( x )$ satisfies $f ( 0 ) > 0$. Define the function $g ( x )$ as
$$g ( x ) = \left| \int _ { 0 } ^ { x } f ( t ) d t \right|$$
The graph of the function $y = g ( x )$ is as shown in the figure. Which of the following statements are correct? Choose all that apply. [4 points]

ㄱ. The equation $f ( x ) = 0$ has three distinct real roots. ㄴ. $f ^ { \prime } ( 0 ) < 0$ ㄷ. The number of natural numbers $m$ satisfying $\int _ { m } ^ { m + 2 } f ( x ) d x > 0$ is 3.
(1) ㄴ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For the function $f ( x ) = k x ^ { 2 } e ^ { - x } ( k > 0 )$ and a real number $t$, let $g ( t )$ be the smaller of the distance from the point $( t , f ( t ) )$ on the curve $y = f ( x )$ to the $x$-axis and the distance to the $y$-axis. What is the maximum value of $k$ such that the function $g ( t )$ is not differentiable at exactly one point? [4 points]
(1) $\frac { 1 } { e }$
(2) $\frac { 1 } { \sqrt { e } }$
(3) $\frac { e } { 2 }$
(4) $\sqrt { e }$
(5) $e$

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]

For the function $f ( x ) = x ( x + 1 ) ( x - 4 )$, answer the following. When the line $y = 5 x + k$ and the graph of the function $y = f ( x )$ intersect at two distinct points, what is the value of the positive number $k$? [4 points]
(1) 5
(2) $\frac { 11 } { 2 }$
(3) 6
(4) $\frac { 13 } { 2 }$
(5) 7

As shown in the figure, there is an isosceles triangle ABC with $\angle \mathrm { CAB } = \angle \mathrm { BCA } = \theta$ that is externally tangent to a circle of radius 1. On the extension of segment AB, a point D (not equal to A) is chosen such that $\angle \mathrm { DCB } = \theta$. Let the area of triangle BDC be $S ( \theta )$. What is the value of $\lim _ { \theta \rightarrow + 0 } \{ \theta \times S ( \theta ) \}$? (Here, $0 < \theta < \frac { \pi } { 4 }$) [4 points]
(1) $\frac { 2 } { 3 }$
(2) $\frac { 8 } { 9 }$
(3) $\frac { 10 } { 9 }$
(4) $\frac { 4 } { 3 }$
(5) $\frac { 14 } { 9 }$

For all cubic functions $f ( x )$ satisfying the following conditions, what is the minimum value of $f ( 2 )$? [4 points] (가) The leading coefficient of $f ( x )$ is 1. (나) $f ( 0 ) = f ^ { \prime } ( 0 )$ (다) For all real numbers $x \geq - 1$, $f ( x ) \geq f ^ { \prime } ( x )$.
(1) 28
(2) 33
(3) 38
(4) 43
(5) 48

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]

In the coordinate plane, point A has coordinates $( 1,0 )$, and for $\theta$ with $0 < \theta < \frac { \pi } { 2 }$, point B has coordinates $( \cos \theta , \sin \theta )$. For point C in the first quadrant such that quadrilateral OACB is a parallelogram, let $f ( \theta )$ be the area of quadrilateral OACB and $g ( \theta )$ be the square of the length of segment OC. What is the maximum value of $f ( \theta ) + g ( \theta )$? (Here, O is the origin.) [4 points]
(1) $2 + \sqrt { 5 }$
(2) $2 + \sqrt { 6 }$
(3) $2 + \sqrt { 7 }$
(4) $2 + 2 \sqrt { 2 }$
(5) 5

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 function $f ( x )$ defined for $x > a$ and a quartic function $g ( x )$ with leading coefficient $-1$ satisfy the following conditions. (Here, $a$ is a constant.)
(a) For all real numbers $x > a$, $$( x - a ) f ( x ) = g ( x ).$$ (b) For two distinct real numbers $\alpha , \beta$, the function $f ( x )$ has the same local maximum value $M$ at $x = \alpha$ and $x = \beta$. (Here, $M > 0$)
(c) The number of values of $x$ where the function $f ( x )$ has a local maximum or minimum is greater than the number of values of $x$ where the function $g ( x )$ has a local maximum or minimum. When $\beta - \alpha = 6 \sqrt { 3 }$, find the minimum value of $M$. [4 points]