13. The graph of the function $f$ shown in the figure above has a vertical tangent at the point $( 2,0 )$ and horizontal tangents at the points $( 1 , - 1 )$ and $( 3,1 )$. For what values of $x , - 2 < x < 4$, is $f$ not differentiable?
(A) 0 only
(B) 0 and 2 only
(C) 1 and 3 only
(D) 0, 1, and 3 only
(E) 0, 1, 2, and 3

Given the function defined by $\mathrm { y } = \mathrm { e } ^ { \sin \mathrm { x } }$ for all x such that $- \pi \leqq \mathrm { x } \leqq 2 \pi$. (a) Find the x - and y -coordinates of all maximum and minimum points on the given interval. Justify your answe (b) On the axes provided, sketch the graph of the function. (c) Write an equation for the axis of symmetry of the graph.

Let $h$ be a function defined for all $x \neq 0$ such that $h(4) = -3$ and the derivative of $h$ is given by $h^{\prime}(x) = \dfrac{x^{2} - 2}{x}$ for all $x \neq 0$.
(a) Find all values of $x$ for which the graph of $h$ has a horizontal tangent, and determine whether $h$ has a local maximum, a local minimum, or neither at each of these values. Justify your answers.
(b) On what intervals, if any, is the graph of $h$ concave up? Justify your answer.
(c) Write an equation for the line tangent to the graph of $h$ at $x = 4$.
(d) Does the line tangent to the graph of $h$ at $x = 4$ lie above or below the graph of $h$ for $x > 4$? Why?

Let $f$ be the function given by $f ( x ) = \frac { \ln x } { x }$ for all $x > 0$. The derivative of $f$ is given by $f ^ { \prime } ( x ) = \frac { 1 - \ln x } { x ^ { 2 } }$.
(a) Write an equation for the line tangent to the graph of $f$ at $x = e ^ { 2 }$.
(b) Find the $x$-coordinate of the critical point of $f$. Determine whether this point is a relative minimum, a relative maximum, or neither for the function $f$. Justify your answer.
(c) The graph of the function $f$ has exactly one point of inflection. Find the $x$-coordinate of this point.
(d) Find $\lim _ { x \rightarrow 0 ^ { + } } f ( x )$.

Consider a differentiable function $f$ having domain all positive real numbers, and for which it is known that $f^{\prime}(x) = (4 - x)x^{-3}$ for $x > 0$.
(a) Find the $x$-coordinate of the critical point of $f$. Determine whether the point is a relative maximum, a relative minimum, or neither for the function $f$. Justify your answer.
(b) Find all intervals on which the graph of $f$ is concave down. Justify your answer.
(c) Given that $f(1) = 2$, determine the function $f$.

If $f ^ { \prime } ( x ) = \sqrt { x ^ { 4 } + 1 } + x ^ { 3 } - 3 x$, then $f$ has a local maximum at $x =$
(A) $-2.314$
(B) $-1.332$
(C) $0.350$
(D) $0.829$
(E) $1.234$

Let $h$ be a function defined for all $x \neq 0$ such that $h(4) = -3$ and the derivative of $h$ is given by $h'(x) = \dfrac{x^2 - 2}{x}$ for all $x \neq 0$.
(a) Find all values of $x$ for which the graph of $h$ has a horizontal tangent, and determine whether $h$ has a local maximum, a local minimum, or neither at each of these values. Justify your answers.
(b) On what intervals, if any, is the graph of $h$ concave up? Justify your answer.
(c) Write an equation for the line tangent to the graph of $h$ at $x = 4$.
(d) Does the line tangent to the graph of $h$ at $x = 4$ lie above or below the graph of $h$ for $x > 4$? Why?

The derivative of a function $f$ is given by $f ^ { \prime } ( x ) = ( x - 3 ) e ^ { x }$ for $x > 0$, and $f ( 1 ) = 7$.
(a) The function $f$ has a critical point at $x = 3$. At this point, does $f$ have a relative minimum, a relative maximum, or neither? Justify your answer.
(b) On what intervals, if any, is the graph of $f$ both decreasing and concave up? Explain your reasoning.
(c) Find the value of $f ( 3 )$.

The function $f$, whose graph is shown above, is defined on the interval $- 2 \leq x \leq 2$. Which of the following statements about $f$ is false?
(A) $f$ is continuous at $x = 0$.
(B) $f$ is differentiable at $x = 0$.
(C) $f$ has a critical point at $x = 0$.
(D) $f$ has an absolute minimum at $x = 0$.
(E) The concavity of the graph of $f$ changes at $x = 0$.

Consider the function $f ( x ) = \frac { 1 } { x ^ { 2 } - k x }$, where $k$ is a nonzero constant. The derivative of $f$ is given by $f ^ { \prime } ( x ) = \frac { k - 2 x } { \left( x ^ { 2 } - k x \right) ^ { 2 } }$.
(a) Let $k = 3$, so that $f ( x ) = \frac { 1 } { x ^ { 2 } - 3 x }$. Write an equation for the line tangent to the graph of $f$ at the point whose $x$-coordinate is 4.
(b) Let $k = 4$, so that $f ( x ) = \frac { 1 } { x ^ { 2 } - 4 x }$. Determine whether $f$ has a relative minimum, a relative maximum, or neither at $x = 2$. Justify your answer.
(c) Find the value of $k$ for which $f$ has a critical point at $x = -5$.
(d) Let $k = 6$, so that $f ( x ) = \frac { 1 } { x ^ { 2 } - 6 x }$. Find the partial fraction decomposition for the function $f$. Find $\int f ( x ) \, dx$.

Let $f ( x ) = ( x - a ) ( x - b ) ^ { 3 } ( x - c ) ^ { 5 } ( x - d ) ^ { 7 }$, where $a , b , c , d$ are real numbers with $a < b < c < d$. Thus $f ( x )$ has 16 real roots counting multiplicities and among them 4 are distinct from each other. Consider $f ^ { \prime } ( x )$, i.e. the derivative of $f ( x )$. Find the following, if you can: (i) the number of real roots of $f ^ { \prime } ( x )$, counting multiplicities, (ii) the number of distinct real roots of $f ^ { \prime } ( x )$.

[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.

8. Let $x$ be a variable that takes real values, and let $f ( x ) = x ^ { 3 } - 3 x$. Which of the following statements is/are true?
(a) $f ( x )$ has a local maximum at $x = - \sqrt { 3 }$
(b) $f ( x )$ has a local maximum at $x = - 1$
(c) $f ( x )$ has a local minimum at $x = \sqrt { 3 }$
(d) $f ( x )$ has a global minimum at $x = 1$

A quartic function $f ( x )$ with positive leading coefficient satisfies the following conditions. $f ^ { \prime } ( x ) = 0$ has three distinct real roots $\alpha , \beta , \gamma ( \alpha < \beta < \gamma )$, and $f ( \alpha ) f ( \beta ) f ( \gamma ) < 0$. Which of the following in are correct? [3 points]
ㄱ. The function $f ( x )$ has a local maximum value at $x = \beta$. ㄴ. The equation $f ( x ) = 0$ has two distinct real roots. ㄷ. If $f ( \alpha ) > 0$, then the equation $f ( x ) = 0$ has a real root less than $\beta$.
(1) ᄀ
(2) ᄃ
(3) ᄀ, ᄂ
(4) ㄴ,ㄷ
(5) ᄀ, ᄂ, ᄃ

The function $f ( x ) = x ^ { 3 } - 12 x$ has a local maximum value $b$ at $x = a$. Find the value of $a + b$. [3 points]

A quartic function $f ( x )$ with leading coefficient 1 satisfies the following conditions. (가) $f ^ { \prime } ( 0 ) = 0 , f ^ { \prime } ( 2 ) = 16$ (나) For some positive number $k$, $f ^ { \prime } ( x ) < 0$ on the two open intervals $( - \infty , 0 ) , ( 0 , k )$. Choose all correct statements from the following. [4 points]
$\langle$Statements$\rangle$ ㄱ. The equation $f ^ { \prime } ( x ) = 0$ has exactly one real root in the open interval $( 0,2 )$. ㄴ. The function $f ( x )$ has a local maximum value. ㄷ. If $f ( 0 ) = 0$, then $f ( x ) \geq - \frac { 1 } { 3 }$ for all real numbers $x$.
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

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 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

10. Given the function $f ( x ) = x ^ { 3 } - x + 1$, then
A. $f ( x )$ has two extreme points
B. $f ( x )$ has three zeros
C. The point $( 0,1 )$ is a center of symmetry of the curve $y = f ( x )$
D. The line $y = 2 x$ is a tangent line to the curve $y = f ( x )$

Given the function $f ( x ) = \left( \frac { 1 } { x } + a \right) \ln ( 1 + x )$.
(1) When $a = - 1$, find the equation of the tangent line to the curve $y = f ( x )$ at the point $( 1 , f ( 1 ) )$.
(2) Do there exist $a$ and $b$ such that the curve $y = f \left( \frac { 1 } { x } \right)$ is symmetric about the line $x = b$? If they exist, find the values of $a$ and $b$. If they do not exist, explain why.
(3) If $f ( x )$ has an extremum on $( 0 , + \infty )$, find the range of $a$.

If the function $f(x)=a\ln x+\frac{b}{x}+\frac{c}{x^2}$ $(a\neq 0)$ has both a local maximum and a local minimum, then:
A. $bc>0$
B. $ab>0$
C. $b^2+8ac>0$
D. $ac<0$

Let $f ( x ) = 2 x ^ { 3 } - 3 a x ^ { 2 } + 1$. Then
A. When $a > 1$, $f ( x )$ has three zeros
B. When $a < 0$, $x = 0$ is a local maximum point of $f ( x )$
C. There exist $a , b$ such that $x = b$ is an axis of symmetry of the curve $y = f ( x )$
D. There exists $a$ such that the point $( 1 , f ( 1 ) )$ is a center of symmetry of the curve $y = f ( x )$

Let function $f ( x ) = ( x - 1 ) ^ { 2 } ( x - 4 )$ , then
A. $x = 3$ is a local minimum point of $f ( x )$
B. When $0 < x < 1$ , $f ( x ) < f \left( x ^ { 2 } \right)$
C. When $1 < x < 2$ , $- 4 < f ( 2 x - 1 ) < 0$
D. When $- 1 < x < 0$ , $f ( 2 - x ) > f ( x )$

(1) [5 marks] Given is the function $f : x \mapsto \frac { e ^ { 2 x } } { x }$ with domain $D _ { f } = \mathbb { R } \backslash \{ 0 \}$. Determine the location and type of the extremum point of the graph of $f$.
Given is the function $f : x \mapsto 1 - \frac { 1 } { x ^ { 2 } }$ defined in $\mathbb { R } \backslash \{ 0 \}$, which has zeros $x _ { 1 } = - 1$ and $x _ { 2 } = 1$. Figure 1 shows the graph of $f$, which is symmetric with respect to the y-axis. Furthermore, the line $g$ with equation $y = - 3$ is given. [Figure]
(2a) [1 marks] Show that one of the points where $g$ intersects the graph of $f$ has the x-coordinate $\frac { 1 } { 2 }$.
(2b) [4 marks] Determine by calculation the area enclosed by the graph of $f$, the x-axis, and the line $g$.
The adjacent Figure 2 shows the graph of a function $f$. [Figure]
(3a) [3 marks] One of the following graphs I, II, and III belongs to the first derivative function of $f$. Specify this graph. Justify why the other two graphs are not suitable. [Figure] [Figure] [Figure]
(3b) [2 marks] The function $F$ is an antiderivative of $f$. Specify the monotonicity behavior of $F$ on the interval $[ 1 ; 3 ]$. Justify your statement.
)$} Consider a family of functions $h _ { k }$ with $k \in \mathbb { R } ^ { + }$, which differ only in their respective domains $D _ { k }$. It holds that $h _ { k } : x \mapsto \cos x$ with $D _ { k } = [ 0 ; k ]$. Figure 4 shows the graph of the function $h _ { 7 }$. Specify the largest possible value of $k$ such that the corresponding function $h _ { k }$ is invertible. For this value of $k$, sketch the graph of the inverse function of $h _ { k }$ in Figure 4 and pay particular attention to the intersection point of the graphs of the function and its inverse. [Figure]
(4b) [2 marks] Specify the term of a function $j$ defined on $\mathbb { R }$ and invertible that satisfies the following condition: The graph of $j$ and the graph of the inverse function of $j$ have no common point.
Given is the function $f : x \mapsto 2 - \ln ( x - 1 )$ with maximal domain $D _ { f }$. The graph of $f$ is denoted by $G _ { f }$.

State the coordinates of the minimum point of the graph of the function $h$ defined in $\mathbb { R }$ with $h ( x ) = - g ( x - 3 )$.
Subtask Part A 4b $( 3 \mathrm { marks } )$ The graph of an antiderivative of $g$ passes through $P$. Sketch this graph in Figure 2.
Given is the function $f : x \mapsto 2 e ^ { - \frac { 1 } { 8 } x ^ { 2 } }$ defined in $\mathbb { R }$. Figure 3 shows the graph $G _ { f }$ of $f$, which has the x-axis as a horizontal asymptote.
[Figure]
Fig. 3
(1a) [2 marks] Calculate the coordinates of the intersection point of $G _ { f }$ with the y-axis and prove by calculation that $G _ { f }$ is symmetric with respect to the y-axis.
(1b) [5 marks] The point $W \left( - 2 \left\lvert \, 2 e ^ { - \frac { 1 } { 2 } } \right. \right)$ is one of the two inflection points of $G _ { f }$. The tangent to $G _ { f }$ at point $W$ is denoted by $w$. Determine an equation of $w$ and calculate the point where $w$ intersects the x-axis. (for verification: $f ^ { \prime } ( x ) = - \frac { 1 } { 2 } x \cdot e ^ { - \frac { 1 } { 8 } x ^ { 2 } }$ )
For each value $c \in \mathbb { R } ^ { + }$, consider the rectangle with vertices $P ( - c \mid 0 ) , Q ( c \mid 0 )$, $R ( c \mid f ( c ) )$ and $S$.
(1c) [1 marks] Draw the rectangle PQRS in Figure 3 for $c = 2$.
(1d) [3 marks] Calculate the value of $c$ for which $\overline { \mathrm { QR } } = 1$ holds.
(1e) [3 marks] State the side lengths of rectangle PQRS as a function of $c$ and justify that the area of the rectangle is given by the term $A ( c ) = 4 c \cdot e ^ { - \frac { 1 } { 8 } c ^ { 2 } }$.
(1f) [4 marks] There is a value of $c$ for which the area $A ( c )$ of rectangle PQRS is maximal. Calculate this value of $c$.
For $k \in \mathbb { R }$, consider the functions $f _ { k } : x \mapsto f ( x ) + k$ defined in $] - \infty ; 0 ]$. Thus $f _ { 0 } ( x ) = f ( x )$, where $f _ { 0 }$ and $f$ differ in their domain.
(1g) [4 marks] Justify using the first derivative of $f _ { k }$ that $f _ { k }$ is invertible for every value of $k$. Sketch the graph of the inverse function of $f _ { 0 }$ in Figure 3.
(1h) [2 marks] State all values of $k$ for which the graph of $f _ { k }$ and the graph of the inverse function of $f _ { k }$ have no common point.
[Figure]
Fig. 4
Figure 4 shows a house with a roof dormer, whose front is shown schematically in Figure 5. The front is described by a model as the region enclosed by the graph $G _ { f }$ of the function $f$ from Part B Subtask 1, the x-axis, and the lines with equations $x = - 4$ and $x = 4$. Here, one unit of length in the coordinate system corresponds to one meter in reality.
[Figure]
Fig. 5
(2a) [2 marks] State the width and height of the front of the roof dormer.
In the front of the roof dormer there is a window. In the model, the window corresponds to the region enclosed by the graph of the function $g$ with $g ( x ) = a x ^ { 4 } + b$ and suitable values $a , b \in \mathbb { R }$ with the x-axis (see Figure 5).
(2b) [2 marks] Justify that $a$ is negative and $b$ is positive.
To determine the area of the front of the roof dormer, an antiderivative $F$ of $f$ is considered.
(2c) [2 marks] One of the graphs I, II and III is the graph of $F$. Justify that this is Graph I by giving one reason each for why Graph II and Graph III do not apply. [Figure] [Figure] [Figure]
(2d) [5 marks] Now determine the area of the entire front of the roof dormer (including the window) using the graph of $F$ from Part B Subtask 2c. Describe, incorporating this area, the essential steps of a solution method by which the value of $a$ could be calculated so that with a window height of 1.50 m, the part of the front of the roof dormer shown shaded in Figure 5 has an area of $6 \mathrm {~m} ^ { 2 }$.
(2e) [5 marks] In order to calculate an approximate value for the length of the upper profile line of the front of the roof dormer, $G _ { f }$ in the range $- 4 \leq x \leq 4$ is approximated by four circular arcs that transition seamlessly into one another and are congruent to each other. One of these circular arcs extends in the range $0 \leq x \leq 2$ and is part of the circle with center $M ( 0 \mid - 1 )$ and radius 3. Calculate the central angle of the circular sector corresponding to this circular arc and use it to determine the desired approximate value.