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 $x = 0$.

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.

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 $- 1 < x < 0$ , $f ( 2 - x ) > f ( x )$

In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$.
By distinguishing cases according to whether $\beta \leqslant 1$ or $\beta > 1$, show that $G_h$ admits a minimum on $\mathbb{R}_+$ attained at a unique point $u_h$. Show moreover that

• [(a)] if $\beta \leqslant 1$ and $h = 0$, then $u_h = 0$;
• [(b)] if $\beta > 1$, then $u_h > 0$;
• [(c)] for any $\beta \in \mathbb{R}_+^*$,
  • [(i)] $G_h'(u_h) = 0$;
  • [(ii)] $h = \beta G_0'(u_h)$;
  • [(iii)] $G_h''(u_h) > 0$ when $h > 0$.

Consider the function $f ( x ) = x ( x - 1 ) e ^ { 2 x }$ if $x \leq 0$ $f ( x ) = x ( 1 - x ) e ^ { - 2 x }$ if $x > 0$ Then $f ( x )$ attains its maximum value at
(a) $1 - 1 / \sqrt{2}$
(b) $1 + 1 / \sqrt{2}$
(c) $- 1 / \sqrt{2}$
(d) $1 / \sqrt{2}$

Show that the function $f ( x )$ defined below attains a unique minimum for $x > 0$. What is the minimum value of the function? What is the value of $x$ at which the minimum is attained? $$f ( x ) = x ^ { 2 } + x + \frac { 1 } { x } + \frac { 1 } { x ^ { 2 } } \quad \text { for } \quad x \neq 0 .$$ Sketch on plain paper the graph of this function.

Find the maximum among $1,2 ^ { 1/2 } , 3 ^ { 1/3 } , 4 ^ { 1/4 } , \ldots$.

Show that the function $f ( x )$ defined below attains a unique minimum for $x > 0$. What is the minimum value of the function? What is the value of $x$ at which the minimum is attained? $$f ( x ) = x ^ { 2 } + x + \frac { 1 } { x } + \frac { 1 } { x ^ { 2 } } \quad \text { for } \quad x \neq 0 .$$ Sketch on plain paper the graph of this function.

Find the maximum among $1,2 ^ { 1/2 } , 3 ^ { 1/3 } , 4 ^ { 1/4 } , \ldots$.

Let $f ( x )$ be a degree 4 polynomial with real coefficients. Let $z$ be the number of real zeroes of $f$, and $e$ be the number of local extrema (i.e., local maxima or minima) of $f$. Which of the following is a possible $( z , e )$ pair?
(A) $( 4,4 )$
(B) $( 3,3 )$
(C) $( 2,2 )$
(D) $( 0,0 )$

Let $f$ be a real-valued differentiable function defined on the real line $\mathbb { R }$ such that its derivative $f ^ { \prime }$ is zero at exactly two distinct real numbers $\alpha$ and $\beta$. Then,
(A) $\alpha$ and $\beta$ are points of local maxima of the function $f$.
(B) $\alpha$ and $\beta$ are points of local minima of the function $f$.
(C) one must be a point of local maximum and the other must be a point of local minimum of $f$.
(D) given data is insufficient to conclude about either of them being local extrema points.