Given is the function $h$ with $h ( x ) = x ^ { 2 } \cdot \mathrm { e } ^ { - x } , x \in \mathbb { R }$.
(1) (i) Show: $h ^ { \prime } ( x ) = \mathrm { e } ^ { - x } \cdot \left( - x ^ { 2 } + 2 x \right)$.
(ii) Calculate the coordinates and the type of local extreme points of the graph of $h$.
[For verification: The local maximum point of $h$ is $x = 2$.]
(2) The points $P ( 0 \mid 0 ) , Q ( r \mid 0 )$ and $R ( r \mid h ( r ) )$ form the vertices of a triangle $P Q R$ for $0 \leq r \leq 10$. Determine $r$ so that the area of triangle $P Q R$ is maximal.
(3) Describe how the graph of $j$ with $j ( x ) = 3 \cdot ( x - 2 ) ^ { 2 } \cdot \mathrm { e } ^ { - ( x - 2 ) }$ is obtained from the graph of $h$. Give the local maximum point of the graph of the function $j$.

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 $h > 0$.

122- The figure below shows the graph of $y = \dfrac{x^2 + ax^2}{x^2 + bx + 1}$. What is the value of the relative minimum of the function?
[Figure: Graph of the function with a local minimum visible, axes labeled $x$ and $y$]
(1) $4/5$
(2) $6$
(3) $6/25$
(4) $6/75$

120. The mean value theorem applies to the function $y = \sqrt{21 - x^2 + 4x}$ on the interval $[6,\ 8]$. For the instantaneous rate of change to equal the average rate of change of this function, what value of $x$ is required?
(1) $4 + \sqrt{7}$ (2) $3 + 2\sqrt{7}$ (3) $2 + \dfrac{3}{2}\sqrt{7}$ (4) $2 + \dfrac{5}{2}\sqrt{7}$

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.

The function $x ^ { 2 } \log _ { e } x$ in the interval $( 0,2 )$ has:
(A) exactly one point of local maximum and no points of local minimum.
(B) exactly one point of local minimum and no points of local maximum.
(C) points of local maximum as well as local minimum.
(D) neither a point of local maximum nor a point of local minimum.

Consider the function $$f(x) = \sum_{k=1}^{m} (x-k)^4, \quad x \in \mathbb{R}$$ where $m > 1$ is an integer. Show that $f$ has a unique minimum and find the point where the minimum is attained.

Suppose $F : \mathbb { R } \rightarrow \mathbb { R }$ is a continuous function which has exactly one local maximum. Then which of the following is true?
(A) $F$ cannot have a local minimum.
(B) $F$ must have exactly one local minimum.
(C) $F$ must have at least two local minima.
(D) $F$ must have either a global maximum or a local minimum.

Suppose that $f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + d$ where $a , b , c , d$ are real numbers with $a \neq 0$. The equation $f ( x ) = 0$ has exactly two distinct real solutions. If $f ^ { \prime } ( x )$ is the derivative of $f ( x )$, then which of the following is a possible graph of $f ^ { \prime } ( x )$?
(A), (B), (C), (D) [graphs as provided in the figure]

Let $f(x) = 2x^3 - 3x^2 - 12x + 4$. Then
(A) $f$ has a local maximum at $x = -1$ and a local minimum at $x = 2$
(B) $f$ has a local minimum at $x = -1$ and a local maximum at $x = 2$
(C) $f$ has local minima at $x = -1$ and at $x = 2$
(D) $f$ has local maxima at $x = -1$ and at $x = 2$

Consider the function $f : ( - \infty , \infty ) \rightarrow ( - \infty , \infty )$ defined by
$$f ( x ) = \frac { x ^ { 2 } - a x + 1 } { x ^ { 2 } + a x + 1 } , 0 < a < 2 .$$
Which of the following is true?
(A) $f ( x )$ is decreasing on $( - 1,1 )$ and has a local minimum at $x = 1$
(B) $f ( x )$ is increasing on $( - 1,1 )$ and has a local maximum at $x = 1$
(C) $f ( x )$ is increasing on $( - 1,1 )$ but has neither a local maximum nor a local minimum at $x = 1$
(D) $f ( x )$ is decreasing on $( - 1,1 )$ but has neither a local maximum nor a local minimum at $x = 1$

Let f be a function defined on $\mathbf { R }$ (the set of all real numbers) such that $\mathrm { f } ^ { \prime } ( \mathrm { x } ) = 2010 ( \mathrm { x } - 2009 ) ( \mathrm { x } - 2010 ) ^ { 2 } ( \mathrm { x } - 2011 ) ^ { 3 } ( \mathrm { x } - 2012 ) ^ { 4 }$, for all $\mathrm { x } \in \mathbf { R }$.
If $g$ is a function defined on $\mathbf { R }$ with values in the interval $( 0 , \infty )$ such that
$$\mathrm { f } ( \mathrm { x } ) = \ell n ( \mathrm {~g} ( \mathrm { x } ) ) \text {, for all } \mathrm { x } \in \mathbf { R } \text {, }$$
then the number of points in $\mathbf { R }$ at which $g$ has a local maximum is

Let $f : \mathbb { R } \rightarrow ( 0 , \infty )$ and $g : \mathbb { R } \rightarrow \mathbb { R }$ be twice differentiable functions such that $f ^ { \prime \prime }$ and $g ^ { \prime \prime }$ are continuous functions on $\mathbb { R }$. Suppose $f ^ { \prime } ( 2 ) = g ( 2 ) = 0 , \quad f ^ { \prime \prime } ( 2 ) \neq 0$ and $g ^ { \prime } ( 2 ) \neq 0$. If $\lim _ { x \rightarrow 2 } \frac { f ( x ) g ( x ) } { f ^ { \prime } ( x ) g ^ { \prime } ( x ) } = 1$, then
(A) $f$ has a local minimum at $x = 2$
(B) $f$ has a local maximum at $x = 2$
(C) $f ^ { \prime \prime } ( 2 ) > f ( 2 )$
(D) $f ( x ) - f ^ { \prime \prime } ( x ) = 0$ for at least one $x \in \mathbb { R }$

Let $$f(x) = \frac{\sin\pi x}{x^2}, \quad x > 0$$
Let $x_1 < x_2 < x_3 < \cdots < x_n < \cdots$ be all the points of local maximum of $f$ and $y_1 < y_2 < y_3 < \cdots < y_n < \cdots$ be all the points of local minimum of $f$. Then which of the following options is/are correct?
(A) $x_1 < y_1$
(B) $x_{n+1} - x_n > 2$ for every $n$
(C) $x_n \in \left(2n, 2n + \frac{1}{2}\right)$ for every $n$
(D) $|x_n - y_n| > 1$ for every $n$

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be given by $f(x) = (x-1)(x-2)(x-5)$. Define $$F(x) = \int_0^x f(t)\,dt, \quad x > 0$$
Then which of the following options is/are correct?
(A) $F$ has a local minimum at $x = 1$
(B) $F$ has a local maximum at $x = 2$
(C) $F$ has two local maxima and one local minimum in $(0, \infty)$
(D) $F(x) \neq 0$ for all $x \in (0, 5)$

Let the function $f: (0, \pi) \rightarrow \mathbb{R}$ be defined by $$f(\theta) = (\sin\theta + \cos\theta)^{2} + (\sin\theta - \cos\theta)^{4}$$ Suppose the function $f$ has a local minimum at $\theta$ precisely when $\theta \in \{\lambda_{1}\pi, \ldots, \lambda_{r}\pi\}$, where $0 < \lambda_{1} < \cdots < \lambda_{r} < 1$. Then the value of $\lambda_{1} + \cdots + \lambda_{r}$ is $\_\_\_\_$

Let $f_1 : (0, \infty) \to \mathbb{R}$ and $f_2 : (0, \infty) \to \mathbb{R}$ be defined by $$f_1(x) = \int_0^x \prod_{j=1}^{21} (t - j)^j \, dt, \quad x > 0$$ and $$f_2(x) = 98(x-1)^{50} - 600(x-1)^{49} + 2450, \quad x > 0,$$ where, for any positive integer $n$ and real numbers $a_1, a_2, \ldots, a_n$, $\prod_{i=1}^n a_i$ denotes the product of $a_1, a_2, \ldots, a_n$. Let $m_i$ and $n_i$, respectively, denote the number of points of local minima and the number of points of local maxima of function $f_i$, $i = 1, 2$, in the interval $(0, \infty)$.
The value of $2m_1 + 3n_1 + m_1 n_1$ is ____.

Let $f_1 : (0, \infty) \to \mathbb{R}$ and $f_2 : (0, \infty) \to \mathbb{R}$ be defined by $$f_1(x) = \int_0^x \prod_{j=1}^{21} (t - j)^j \, dt, \quad x > 0$$ and $$f_2(x) = 98(x-1)^{50} - 600(x-1)^{49} + 2450, \quad x > 0,$$ where, for any positive integer $n$ and real numbers $a_1, a_2, \ldots, a_n$, $\prod_{i=1}^n a_i$ denotes the product of $a_1, a_2, \ldots, a_n$. Let $m_i$ and $n_i$, respectively, denote the number of points of local minima and the number of points of local maxima of function $f_i$, $i = 1, 2$, in the interval $(0, \infty)$.
The value of $6m_2 + 4n_2 + 8m_2 n_2$ is ____.

Let $f _ { 1 } : ( 0 , \infty ) \rightarrow \mathbb { R }$ and $f _ { 2 } : ( 0 , \infty ) \rightarrow \mathbb { R }$ be defined by $$f _ { 1 } ( x ) = \int _ { 0 } ^ { x } \prod _ { j = 1 } ^ { 21 } ( t - j ) ^ { j } d t , \quad x > 0$$ and $$f _ { 2 } ( x ) = 98 ( x - 1 ) ^ { 50 } - 600 ( x - 1 ) ^ { 49 } + 2450 , \quad x > 0$$ where, for any positive integer $n$ and real numbers $a _ { 1 } , a _ { 2 } , \ldots , a _ { n } , \prod _ { i = 1 } ^ { n } a _ { i }$ denotes the product of $a _ { 1 } , a _ { 2 } , \ldots , a _ { n }$. Let $m _ { i }$ and $n _ { i }$, respectively, denote the number of points of local minima and the number of points of local maxima of function $f _ { i } , i = 1,2$, in the interval $( 0 , \infty )$. The value of $2 m _ { 1 } + 3 n _ { 1 } + m _ { 1 } n _ { 1 }$ is $\_\_\_\_$.