Let $f$ be the function given by $f(x) = 2xe^{2x}$.
(a) Find $\lim_{x \rightarrow -\infty} f(x)$ and $\lim_{x \rightarrow \infty} f(x)$.
(b) Find the absolute minimum value of $f$. Justify that your answer is an absolute minimum.
(c) What is the range of $f$?
(d) Consider the family of functions defined by $y = bxe^{bx}$, where $b$ is a nonzero constant. Show that the absolute minimum value of $bxe^{bx}$ is the same for all nonzero values of $b$.

The number of gallons, $P(t)$, of a pollutant in a lake changes at the rate $P^{\prime}(t) = 1 - 3e^{-0.2\sqrt{t}}$ gallons per day, where $t$ is measured in days. There are 50 gallons of the pollutant in the lake at time $t = 0$. The lake is considered to be safe when it contains 40 gallons or less of pollutant.
(a) Is the amount of pollutant increasing at time $t = 9$? Why or why not?
(b) For what value of $t$ will the number of gallons of pollutant be at its minimum? Justify your answer.
(c) Is the lake safe when the number of gallons of pollutant is at its minimum? Justify your answer.
(d) An investigator uses the tangent line approximation to $P(t)$ at $t = 0$ as a model for the amount of pollutant in the lake. At what time $t$ does this model predict that the lake becomes safe?

Let $f$ be the function defined by $f ( x ) = \frac { \ln x } { x }$. What is the absolute maximum value of $f$ ?
(A) 1
(B) $\frac { 1 } { e }$
(C) 0
(D) $- e$
(E) $f$ does not have an absolute maximum value.

Let $f$ be the function given by $f(x) = 2xe^{2x}$.
(a) Find $\lim_{x \rightarrow -\infty} f(x)$ and $\lim_{x \rightarrow \infty} f(x)$.
(b) Find the absolute minimum value of $f$. Justify that your answer is an absolute minimum.
(c) What is the range of $f$?
(d) Consider the family of functions defined by $y = bxe^{bx}$, where $b$ is a nonzero constant. Show that the absolute minimum value of $bxe^{bx}$ is the same for all nonzero values of $b$.

The rate at which people enter an auditorium for a rock concert is modeled by the function $R$ given by $R(t) = 1380t^{2} - 675t^{3}$ for $0 \leq t \leq 2$ hours; $R(t)$ is measured in people per hour. No one is in the auditorium at time $t = 0$, when the doors open. The doors close and the concert begins at time $t = 2$.
(a) How many people are in the auditorium when the concert begins?
(b) Find the time when the rate at which people enter the auditorium is a maximum. Justify your answer.
(c) The total wait time for all the people in the auditorium is found by adding the time each person waits, starting at the time the person enters the auditorium and ending when the concert begins. The function $w$ models the total wait time for all the people who enter the auditorium before time $t$. The derivative of $w$ is given by $w'(t) = (2 - t)R(t)$. Find $w(2) - w(1)$, the total wait time for those who enter the auditorium after time $t = 1$.
(d) On average, how long does a person wait in the auditorium for the concert to begin? Consider all people who enter the auditorium after the doors open, and use the model for total wait time from part (c).

For each real number $a$, we consider the function $f _ { a }$ defined on the set of real numbers $\mathbb { R }$ by
$$f _ { a } ( x ) = \mathrm { e } ^ { x - a } - 2 x + \mathrm { e } ^ { a } .$$

1. Show that for every real number $a$, the function $f _ { a }$ has a minimum.
2. Does there exist a value of $a$ for which this minimum is as small as possible?

The rate (as a percentage) of $\mathrm{CO}_2$ contained in a room after $t$ minutes of hood operation is modelled by the function $f$ defined for all real $t$ in the interval $[0;20]$ by: $$f(t) = (0{,}8t + 0{,}2)\mathrm{e}^{-0{,}5t} + 0{,}03.$$ In this question, round both results to the nearest thousandth. a. Calculate $f(20)$. b. Determine the maximum rate of $\mathrm{CO}_2$ present in the room during the experiment.

Exercise 1 (5 points)
The plane is equipped with an orthogonal coordinate system $(\mathrm{O}, \mathrm{I}, \mathrm{J})$.
It is admitted that, for all real $a$ in the interval $]0; 1]$, the area of triangle $\mathrm{O}N_aP_a$ in square units is given by $\mathscr{A}(a) = \frac{1}{2}a(1 - \ln a)^2$.
Using the previous questions, determine for which value of $a$ the area $\mathscr{A}(a)$ is maximum. Determine this maximum area.

Part A
Let $g$ be the function defined on the set of real numbers $\mathbf { R }$, by $$g ( x ) = x ^ { 2 } + x + \frac { 1 } { 4 } + \frac { 4 } { \left( 1 + \mathrm { e } ^ { x } \right) ^ { 2 } }$$
It is admitted that the function $g$ is differentiable on $\mathbf { R }$ and we denote by $g ^ { \prime }$ its derivative function.
1. Determine the limits of $g$ at $+ \infty$ and at $- \infty$.
2. It is admitted that the function $g ^ { \prime }$ is strictly increasing on $\mathbf { R }$ and that $g ^ { \prime } ( 0 ) = 0$.
Determine the sign of the function $g ^ { \prime }$ on $\mathbf { R }$.
3. Draw up the table of variations of the function $g$ and calculate the minimum of the function $g$ on $\mathbf { R }$.
Part B
Let $f$ be the function defined on $\mathbf { R }$ by: $$f ( x ) = 3 - \frac { 2 } { 1 + \mathrm { e } ^ { x } }$$
We denote by $\mathscr { C } _ { f }$ the representative curve of $f$ in an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath }$ ).
Let A be the point with coordinates $\left( - \frac { 1 } { 2 } ; 3 \right)$.
1. Prove that point $\mathrm { B } ( 0 ; 2 )$ belongs to $\mathscr { C } _ { f }$.
2. Let $x$ be any real number. We denote by $M$ the point on the curve $\mathscr { C } _ { f }$ with coordinates $( x ; f ( x ) )$.
Prove that $\mathrm { A } M ^ { 2 } = g ( x )$.
3. It is admitted that the distance $\mathrm { A } M$ is minimal if and only if $\mathrm { A } M ^ { 2 }$ is minimal.
Determine the coordinates of the point on the curve $\mathscr { C } _ { f }$ such that the distance AM is minimal.
4. It is admitted that the function $f$ is differentiable on $\mathbf { R }$ and we denote by $f ^ { \prime }$ its derivative function.
a. Calculate $f ^ { \prime } ( x )$ for all real $x$.
b. Let $T$ be the tangent to the curve $\mathscr { C } _ { f }$ at point B.
Prove that the reduced equation of $T$ is $y = \frac { x } { 2 } + 2$.
5. Prove that the line $T$ is perpendicular to the line (AB).

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

For all real numbers $x$ with $- 2 \leq x \leq 2$, the inequality $$- k \leq 2 x ^ { 3 } + 3 x ^ { 2 } - 12 x - 8 \leq k$$ holds. Find the minimum value of the positive number $k$. [3 points]

17. Let a be a real number. The maximum value of the function $f ( x ) = \left| x ^ { 2 } - a x \right|$ on the interval $[ 0,1 ]$ is denoted by $g ( a )$. When $a =$ $\_\_\_\_$,
$$\mathbf { y }$$
the value of $g ( a )$ is minimized. III. Solution Questions

Given the function $f ( x ) = \frac { 1 } { 4 } x ^ { 3 } - x ^ { 2 } + x$. (I) Find the equation of the tangent line to the curve $y = f ( x )$ with slope 1; (II) When $x \in [ - 2,4 ]$, prove that: $x - 6 \leqslant f ( x ) \leqslant x$; (III) Let $F ( x ) = | f ( x ) - ( x + a ) | ( a \in \mathbb { R } )$. Let $M ( a )$ denote the maximum value of $F ( x )$ on the interval $[ - 2,4 ]$. When $M ( a )$ is minimized, find the value of $a$.

The function $f ( x ) = \cos x + ( x + 1 ) \sin x + 1$ on the interval $[ 0,2 \pi ]$ has minimum and maximum values respectively
A. $- \frac { \pi } { 2 } , \frac { \pi } { 2 }$
B. $- \frac { 3 \pi } { 2 } , \frac { \pi } { 2 }$
C. $- \frac { \pi } { 2 } , \frac { \pi } { 2 } + 2$
D. $- \frac { 3 \pi } { 2 } , \frac { \pi } { 2 } + 2$

Determine $\max_{1 \leqslant p \leqslant n-1} \left( n^{2} - pn + p^{2} \right)$.

In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$.
We now assume that $h > 0$. Justify that $G_h$ attains its minimum on $\mathbb{R}$ at a unique point which is $u_h$.

The minimum value of $x_1^{2} + x_2^{2} + x_3^{2} + x_4^{2}$ subject to $x_1 + x_2 + x_3 + x_4 = a$ and $x_1 - x_2 + x_3 - x_4 = b$ is
(a) $(a^{2} + b^{2})/4$
(b) $(a^{2} + b^{2})/2$
(c) $(a+b)^{2}/4$
(d) $(a+b)^{2}/2$

Let $f(x) = \dfrac{2x^2 + 3x + 1}{2x - 1}$. Find the maximum and minimum values of $f$ on $[2, 3]$.

If $xy = 1$, find the minimum value of $\dfrac{4}{4-x^2} + \dfrac{9}{9-y^2}$.

Find the maximum value of $x ^ { 2 } + y ^ { 2 }$ in the bounded region, including the boundary, enclosed by $y = \frac { x } { 2 } , y = - \frac { x } { 2 }$ and $x = y ^ { 2 } + 1$.

Find the maximum value of $x ^ { 2 } + y ^ { 2 }$ in the bounded region, including the boundary, enclosed by $y = \frac { x } { 2 } , y = - \frac { x } { 2 }$ and $x = y ^ { 2 } + 1$.

For a positive real number $\alpha$, let $S_\alpha$ denote the set of points $(x, y)$ satisfying $$|x|^\alpha + |y|^\alpha = 1$$ A positive number $\alpha$ is said to be good if the points in $S_\alpha$ that are closest to the origin lie only on the coordinate axes. Then
(A) all $\alpha$ in $(0,1)$ are good and others are not good.
(B) all $\alpha$ in $(1,2)$ are good and others are not good.
(C) all $\alpha > 2$ are good and others are not good.
(D) all $\alpha > 1$ are good and others are not good.

Let $S = \left\{ x - y \mid x , y \text{ are real numbers with } x ^ { 2 } + y ^ { 2 } = 1 \right\}$. Then the maximum number in the set $S$ is
(A) 1
(B) $\sqrt { 2 }$
(C) $2 \sqrt { 2 }$
(D) $1 + \sqrt { 2 }$.

If $x , y$ are positive real numbers such that $3 x + 4 y < 72$, then the maximum possible value of $12 x y ( 72 - 3 x - 4 y )$ is:
(A) 12240
(B) 13824
(C) 10656
(D) 8640

The maximum value of the function $f(x)=2x^{3}-15x^{2}+36x-48$ on the set $A=\left\{x\mid x^{2}+20\leq9x\right\}$ is