Question 144
A figura mostra o gráfico de uma função $f$.
[Figure]
Com base no gráfico, é correto afirmar que
(A) $f(-2) = 0$ (B) $f(0) = -2$ (C) $f(1) = 3$ (D) $f(2) = 0$ (E) $f(3) = 1$

A pharmaceutical company conducted a study of the efficacy (in percentage) of a medication over 12 hours of treatment in a patient. The medication was administered in two doses, with a 6-hour interval between them. As soon as the first dose was administered, the medication's efficacy increased linearly for 1 hour, until reaching maximum efficacy (100\%), and remained at maximum efficacy for 2 hours. After these 2 hours at maximum efficacy, it began to decrease linearly, reaching 20\% efficacy upon completing the initial 6 hours of analysis. At this moment, the second dose was administered, which began to increase linearly, reaching maximum efficacy after 0.5 hours and remaining at 100\% for 3.5 hours. In the remaining hours of analysis, the efficacy decreased linearly, reaching 50\% efficacy at the end of treatment.
Considering the quantities time (in hours) on the horizontal axis and medication efficacy (in percentage) on the vertical axis, which graph represents this study?
(A), (B), (C), (D), (E) [see figures]

Let $$f ( x ) = \frac { x ^ { 4 } } { ( x - 1 ) ( x - 2 ) \cdots ( x - n ) }$$ where the denominator is a product of $n$ factors, $n$ being a positive integer. It is also given that the $X$-axis is a horizontal asymptote for the graph of $f$. Answer the independent questions below by choosing the correct option from the given ones. a) How many vertical asymptotes does the graph of $f$ have?
Options: $n$ less than $n$ more than $n$ impossible to decide
Answer: $\_\_\_\_$ b) What can you deduce about the value of $n$?
Options: $n < 4$ $n = 4$ $n > 4$ impossible to decide
Answer: $\_\_\_\_$ c) As one travels along the graph of $f$ from left to right, at which of the following points is the sign of $f ( x )$ guaranteed to change from positive to negative?
Options: $x = 0$ $x = 1$ $x = n - 1$ $x = n$
Answer: $\_\_\_\_$ d) How many inflection points does the graph of $f$ have in the region $x < 0$?
Options: none 1 more than 1 impossible to decide
(Hint: Sketching is better than calculating.)
Answer: $\_\_\_\_$

Write your answers to each question below as a series of three letters Y (for Yes) or N (for No). Leave space between the group of three letters answering (i), the answers to (ii) and the answers to (iii). Consider the graphs of functions $$f(x) = \frac{x^{3}}{x^{2}-x} \qquad g(x) = \frac{x^{2}-x}{x^{3}} \qquad h(x) = \frac{x^{3}-x}{x^{3}+x}$$ (i) Does $f$ have a horizontal asymptote? A vertical asymptote? A removable discontinuity?
(ii) Does $g$ have a horizontal asymptote? A vertical asymptote? A removable discontinuity?
(iii) Does $h$ have a horizontal asymptote? A vertical asymptote? A removable discontinuity?

Consider the following conditions on a function $f$ whose domain is the closed interval $[0,1]$. (For any condition involving a limit, at the endpoints, use the relevant one-sided limit.) I. $f$ is differentiable at each $x \in [0,1]$. II. $f$ is continuous at each $x \in [0,1]$. III. The set $\{f(x) \mid x \in [0,1]\}$ has a maximum element and a minimum element.
Statements
(13) If I is true, then II is true. (14) If II is true, then III is true. (15) If III is false, then I is false. (16) No two of the three given conditions are equivalent to each other. (Two statements being equivalent means each implies the other.)

Let $$f(x) = \frac{1}{|\ln x|}\left(\frac{1}{x} + \cos x\right)$$
Statements
(21) As $x \rightarrow \infty$, the sign of $f(x)$ changes infinitely many times. (22) As $x \rightarrow \infty$, the limit of $f(x)$ does not exist. (23) As $x \rightarrow 1$, $f(x) \rightarrow \infty$. (24) As $x \rightarrow 0^+$, $f(x) \rightarrow 1$.

Statements
(13) As $x \rightarrow - \infty$ the function $\cos \left( e ^ { x } \right)$ tends to a finite limit. (14) As $x \rightarrow \infty$ the function $\cos \left( e ^ { x } \right)$ changes sign infinitely many times. (15) As $x \rightarrow \infty$, the function $\sin ( \ln ( x ) )$ tends to a finite limit. (16) $\sin ( \ln ( x ) )$ changes sign only finitely many times as $x$ goes towards 0 from 1.

Consider the part of the graph of $y ^ { 2 } + x ^ { 3 } = 15 x y$ that is strictly to the right of the $Y$-axis, i.e., take only the points on the graph with $x > 0$.
Questions
(33) Write the least possible value of $y$ among considered points. If there is no such real number, write NONE. (34) Write the largest possible value of $y$ among considered points. If there is no such real number, write NONE.

(a) Draw a qualitatively accurate sketch of the unique bounded region R in the first quadrant that has maximum possible finite area with boundary described as follows. R is bounded below by the graph of $y = x^2 - x^3$, bounded above by the graph of an equation of the form $y = kx$ (where $k$ is some constant), and R is entirely enclosed by the two given graphs, i.e., the boundary of the region R must be a subset of the union of the given two graphs (so R does not have any points on its boundary that are not on these two graphs). Clearly mark the relevant point(s) on the boundary where the two given graphs meet and write the coordinates of every such point.
(b) Consider the solid obtained by rotating the above region R around $Y$-axis. Show how to find the volume of this solid by doing the following: Carefully set up the calculation with justification. Do enough work with the resulting expression to reach a stage where the final numerical answer can be found mechanically by using standard symbolic formulas of algebra and/or calculus and substituting known values in them. Do not carry out the mechanical work to get the final numerical answer.

11. Let $x$ be a real-valued variable. What is the range of the function $f ( x ) = \sin ^ { 2 } x - \sin x + 2$ ?
(a) $[ 0,2 ]$
(b) $[ 1,2 ]$
(c) $[ 1,4 ]$
(d) $\left[ \frac { 7 } { 4 } , 4 \right]$

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 figure on the right shows 6 semicircles with center at $( 1,1 )$ and radii of lengths $\frac { 1 } { 3 } , \frac { 2 } { 3 } , 1 , \frac { 4 } { 3 } , \frac { 5 } { 3 } , 2$ respectively. Three functions $$\begin{aligned} & y = \log _ { \frac { 1 } { 4 } } x \\ & y = \left( \frac { 2 } { 3 } \right) ^ { x } \\ & y = 3 ^ { x } \end{aligned}$$ Let $a , b , c$ be the number of intersection points where the graphs of these functions meet the semicircles, respectively. Which of the following correctly represents the relationship between $a , b , c$? (Here, $x \geqq 1$ and the semicircles include the endpoints of the diameter.) [4 points]
(1) $a < b < c$
(2) $a < c < b$
(3) $b < c < a$
(4) $c < a < b$
(5) $c < b < a$

For a positive number $a$, on the closed interval $[ - a , a ]$, the function
$$f ( x ) = \frac { x - 5 } { ( x - 5 ) ^ { 2 } + 36 }$$
has maximum value $M$ and minimum value $m$. Find the minimum value of $a$ such that $M + m = 0$. [4 points]

In the coordinate plane, let $C$ be a circle with center $(0,3)$ and radius 1. For a positive number $r$, let $f ( r )$ be the number of circles with radius $r$ that meet circle $C$ at exactly one point and are tangent to the $x$-axis. Which of the following statements in are correct? [4 points]
Remarks ㄱ. $f ( 2 ) = 3$ ㄴ. $\lim _ { r \rightarrow 1 + 0 } f ( r ) = f ( 1 )$ ㄷ. The number of discontinuity points of the function $f ( r )$ on the interval $( 0,4 )$ is 2.
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ

In the coordinate plane, for a natural number $n$, let $A _ { n }$ be a square with vertices at the four points $$\left( n ^ { 2 } , n ^ { 2 } \right) , \left( 4 n ^ { 2 } , n ^ { 2 } \right) , \left( 4 n ^ { 2 } , 4 n ^ { 2 } \right) , \left( n ^ { 2 } , 4 n ^ { 2 } \right)$$ Let $a _ { n }$ be the number of natural numbers $k$ such that the square $A _ { n }$ and the graph of the function $y = k \sqrt { x }$ intersect. Which of the following statements in the given options are correct? [4 points] Given Options ㄱ. $a _ { 5 } = 15$ ㄴ. $a _ { n + 2 } - a _ { n } = 7$ ㄷ. $\sum _ { k = 1 } ^ { 10 } a _ { k } = 200$
(1) ㄴ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

In the coordinate plane, for a natural number $n$, let $A _ { n }$ be a square with vertices at the four points $$\left( n ^ { 2 } , n ^ { 2 } \right) , \left( 4 n ^ { 2 } , n ^ { 2 } \right) , \left( 4 n ^ { 2 } , 4 n ^ { 2 } \right) , \left( n ^ { 2 } , 4 n ^ { 2 } \right)$$ Let $a _ { n }$ be the number of natural numbers $k$ such that the square $A _ { n }$ and the graph of the function $y = k \sqrt { x }$ intersect. Which of the following statements in are correct? [4 points]
Remarks ㄱ. $a _ { 5 } = 15$ ㄴ. $a _ { n + 2 } - a _ { n } = 7$ ㄷ. $\sum _ { k = 1 } ^ { 10 } a _ { k } = 200$
(1) ㄴ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

The graph of the function $y = f ( x )$ defined on the open interval $( - 2,2 )$ is shown in the following figure.
On the open interval $( - 2,2 )$, define the function $g ( x )$ as $$g ( x ) = f ( x ) + f ( - x )$$ Which of the following in are correct? [4 points]
ㄱ. $\lim _ { x \rightarrow 0 } f ( x )$ exists. ㄴ. $\lim _ { x \rightarrow 0 } g ( x )$ exists. ㄷ. The function $g ( x )$ is continuous at $x = 1$.
(1) ᄂ
(2) ᄃ
(3) ᄀ, ᄂ
(4) ᄀ, ᄃ
(5) ᄂ, ᄃ

The figure on the right shows a circle with center at the origin O and radius 1, and the graph of a quadratic function $y = f ( x )$ passing through the point $( 0 , - 1 )$ on the coordinate plane. The equation $$\frac { 1 } { f ( x ) + 1 } - \frac { 1 } { f ( x ) - 1 } = \frac { 2 } { x ^ { 2 } }$$ has how many distinct real roots $x$? [3 points]
(1) 2
(2) 3
(3) 4
(4) 5
(5) 6

As shown in the figure, the graph of the cubic function $y = f ( x )$ is tangent to the $x$-axis at point $\mathrm { P } ( 2,0 )$ and meets the graph of the linear function $y = g ( x )$ only at point P. When $1 < f ( 0 ) < g ( 0 )$, what is the number of real roots of the equation $$f ( x ) + g ( x ) = \frac { 1 } { f ( x ) } + \frac { 1 } { g ( x ) }$$ ? [4 points]
(1) 7
(2) 6
(3) 5
(4) 4
(5) 3

For a natural number $n ( n \geqq 2 )$, let the $x$-coordinates of the two distinct points where the line $y = - x + n$ and the curve $y = \left| \log _ { 2 } x \right|$ meet be $a _ { n }$ and $b _ { n }$ respectively ($a _ { n } < b _ { n }$). Which of the following statements in are correct? [4 points]
ㄱ. $a _ { 2 } < \frac { 1 } { 4 }$ ㄴ. $0 < \frac { a _ { n + 1 } } { a _ { n } } < 1$ ㄷ. $1 - \frac { \log _ { 2 } n } { n } < \frac { b _ { n } } { n } < 1$
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For a natural number $n ( n \geqq 2 )$, let $a _ { n }$ and $b _ { n } \left( a _ { n } < b _ { n } \right)$ be the $x$-coordinates of the two distinct points where the line $y = - x + n$ and the curve $y = \left| \log _ { 2 } x \right|$ meet. Which of the following statements in are correct? [4 points]
Remarks ㄱ. $a _ { 2 } < \frac { 1 } { 4 }$ ㄴ. $0 < \frac { a _ { n + 1 } } { a _ { n } } < 1$ ㄷ. $1 - \frac { \log _ { 2 } n } { n } < \frac { b _ { n } } { n } < 1$
(1) ᄀ
(2) ᄂ
(3) ᄃ
(4) ᄂ, ᄃ
(5) ᄀ, ᄂ, ᄃ

For a quartic function $f ( x )$ with leading coefficient 1, the function $g ( x )$ satisfies the following conditions. (가) When $- 1 \leqq x < 1$, $g ( x ) = f ( x )$. (나) For all real numbers $x$, $g ( x + 2 ) = g ( x )$.
Which of the following statements in are correct? [4 points]
Remarks ㄱ. If $f ( - 1 ) = f ( 1 )$ and $f ^ { \prime } ( - 1 ) = f ^ { \prime } ( 1 )$, then $g ( x )$ is differentiable on the entire set of real numbers. ㄴ. If $g ( x )$ is differentiable on the entire set of real numbers, then $f ^ { \prime } ( 0 ) f ^ { \prime } ( 1 ) < 0$. ㄷ. If $g ( x )$ is differentiable on the entire set of real numbers and $f ^ { \prime } ( 1 ) > 0$, then there exists $c$ in the interval $( - \infty , - 1 )$ such that $f ^ { \prime } ( c ) = 0$.
(1) ᄀ
(2) ᄂ
(3) ᄀ, ᄃ
(4) ㄴ,ㄷ
(5) ᄀ, ᄂ, ᄃ

For the function $$f ( x ) = \begin{cases} x + 2 & ( x < - 1 ) \\ 0 & ( x = - 1 ) \\ x ^ { 2 } & ( - 1 < x < 1 ) \\ x - 2 & ( x \geqq 1 ) \end{cases}$$ which of the following are correct? Choose all that apply from $\langle$Remarks$\rangle$. [3 points]
$\langle$Remarks$\rangle$ ㄱ. $\lim _ { x \rightarrow 1 + 0 } \{ f ( x ) + f ( - x ) \} = 0$ ㄴ. The function $f ( x ) - | f ( x ) |$ is discontinuous at 1 point. ㄷ. There is no constant $a$ such that the function $f ( x ) f ( x - a )$ is continuous on the entire set of real numbers.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

On the coordinate plane, the two points where the two curves $y = \left| \log _ { 2 } x \right|$ and $y = \left( \frac { 1 } { 2 } \right) ^ { x }$ meet are $\mathrm { P } \left( x _ { 1 } , y _ { 1 } \right) , \mathrm { Q } \left( x _ { 2 } , y _ { 2 } \right) \left( x _ { 1 } < x _ { 2 } \right)$, and the point where the two curves $y = \left| \log _ { 2 } x \right|$ and $y = 2 ^ { x }$ meet is $\mathrm { R } \left( x _ { 3 } , y _ { 3 } \right)$. Which of the following statements in are correct? [4 points]
ㄱ. $\frac { 1 } { 2 } < x _ { 1 } < 1$ ㄴ. $x _ { 2 } y _ { 2 } - x _ { 3 } y _ { 3 } = 0$ ㄷ. $x _ { 2 } \left( x _ { 1 } - 1 \right) > y _ { 1 } \left( y _ { 2 } - 1 \right)$
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

In the coordinate plane, let the two points where the curves $y = \left| \log _ { 2 } x \right|$ and $y = \left( \frac { 1 } { 2 } \right) ^ { x }$ meet be $\mathrm { P } \left( x _ { 1 } , y _ { 1 } \right) , \mathrm { Q } \left( x _ { 2 } , y _ { 2 } \right) \left( x _ { 1 } < x _ { 2 } \right)$, and let the point where the curves $y = \left| \log _ { 2 } x \right|$ and $y = 2 ^ { x }$ meet be $\mathrm { R } \left( x _ { 3 } , y _ { 3 } \right)$. Which of the following are correct? Choose all that apply from $\langle$Remarks$\rangle$. [4 points]
$\langle$Remarks$\rangle$ ㄱ. $\frac { 1 } { 2 } < x _ { 1 } < 1$ ㄴ. $x _ { 2 } y _ { 2 } - x _ { 3 } y _ { 3 } = 0$ ㄷ. $x _ { 2 } \left( x _ { 1 } - 1 \right) > y _ { 1 } \left( y _ { 2 } - 1 \right)$
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ