If $f ^ { \prime } ( x ) > 0$ for all real numbers $x$ and $\int _ { 4 } ^ { 7 } f ( t ) d t = 0$, which of the following could be a table of values for the function $f$ ?
(A)

$x$	$f ( x )$
4	$-4$
5	$-3$
7	0

(B)

$x$	$f ( x )$
4	$-4$
5	$-2$
7	5

(C)

$x$	$f ( x )$
4	$-4$
5	6
7	3

(D)

$x$	$f ( x )$
4	0
5	0
7	0

(E)

$x$	$f ( x )$
4	0
5	4
7	6

Pharmacokinetics studies the evolution of a drug after its administration in the body, by measuring its plasma concentration, that is to say its concentration in the plasma. In this exercise we study the evolution of plasma concentration in a patient of the same dose of drug, considering different modes of administration.
Part A: administration by intravenous route
We denote $f ( t )$ the plasma concentration, expressed in microgram per litre ( $\mu \mathrm { g } . \mathrm { L } ^ { - 1 }$ ), of the drug, after $t$ hours following administration by intravenous route. The mathematical model is: $f ( t ) = 20 \mathrm { e } ^ { - 0,1 t }$, with $t \in [ 0 ; + \infty [$. The initial plasma concentration of the drug is therefore $f ( 0 ) = 20 \mu \mathrm {~g} . \mathrm { L } ^ { - 1 }$.

1. The half-life of the drug is the duration (in hours) after which the plasma concentration of the drug is equal to half the initial concentration. Determine this half-life, denoted $t _ { 0,5 }$.
2. It is estimated that the drug is eliminated as soon as the plasma concentration is less than $0.2 \mu \mathrm {~g} . \mathrm { L } ^ { - 1 }$. Determine the time from which the drug is eliminated. The result will be given rounded to the nearest tenth.
3. In pharmacokinetics, we call AUC (or ``area under the curve''), in $\mu \mathrm { g } . \mathrm { L } ^ { - 1 }$, the number $\lim _ { x \rightarrow + \infty } \int _ { 0 } ^ { x } f ( t ) \mathrm { d } t$. Verify that for this model, the AUC is equal to $200 \mu \mathrm {~g} . \mathrm { L } ^ { - 1 }$.

Part B: administration by oral route
We denote $g ( t )$ the plasma concentration of the drug, expressed in microgram per litre ( $\mu g.L^{-1}$ ), after $t$ hours following ingestion by oral route. The mathematical model is: $g ( t ) = 20 \left( \mathrm { e } ^ { - 0,1 t } - \mathrm { e } ^ { - t } \right)$, with $t \in [ 0 ; + \infty [$. In this case, the effect of the drug is delayed, since the initial plasma concentration is equal to: $g ( 0 ) = 0 \mu g . \mathrm { L } ^ { - 1 }$.

1. Prove that, for all $t$ in the interval $[ 0 ; + \infty [$, we have: $g ^ { \prime } ( t ) = 20 \mathrm { e } ^ { - t } \left( 1 - 0,1 \mathrm { e } ^ { 0,9 t } \right)$.
2. Study the variations of the function $g$ on the interval $[ 0 ; + \infty [$. (The limit at $+ \infty$ is not required.) Deduce the duration after which the plasma concentration of the drug is maximum. The result will be given to the nearest minute.

Part C: repeated administration by intravenous route
We decide to inject at regular time intervals the same dose of drug by intravenous route. The time interval (in hours) between two injections is chosen equal to the half-life of the drug, that is to say the number $t _ { 0,5 }$ which was calculated in A - 1. Each new injection causes an increase in plasma concentration of $20 \mu \mathrm {~g} . \mathrm { L } ^ { - 1 }$. We denote $u _ { n }$ the plasma concentration of the drug immediately after the $n$-th injection. Thus, $u _ { 1 } = 20$ and, for all integer $n$ greater than or equal to 1, we have: $u _ { n + 1 } = 0,5 u _ { n } + 20$.

1. Prove by induction that, for all integer $n \geqslant 1 : u _ { n } = 40 - 40 \times 0,5 ^ { n }$.
2. Determine the limit of the sequence $( u _ { n } )$ as $n$ tends to $+ \infty$.
3. We consider that equilibrium is reached as soon as the plasma concentration exceeds $38 \mu \mathrm {~g} . \mathrm { L } ^ { - 1 }$. Determine the minimum number of injections necessary to reach this equilibrium.

A treatment protocol for a disease in children involves long-term infusion of an appropriate medication. The concentration of the medication in the blood over time is modeled by the function $C$ defined on the interval $[0; +\infty[$ by:
$$C ( t ) = \frac { d } { a } \left( 1 - \mathrm { e } ^ { - \frac { a } { 80 } t } \right)$$
The clearance $a$ of a certain patient is 7, and we choose an infusion rate $d$ equal to 84. In this part, the function $C$ is therefore defined on $[0; +\infty[$ by:
$$C ( t ) = 12 \left( 1 - \mathrm { e } ^ { - \frac { 7 } { 80 } t } \right)$$

1. Study the monotonicity of the function $C$ on $[0; +\infty[$.
2. For the treatment to be effective, the plateau must equal 15. Is the treatment of this patient effective?

Exercise 4 — Candidates who have not followed the specialization course
We are interested in the fall of a water droplet that detaches from a cloud without initial velocity. A very simplified model makes it possible to establish that the instantaneous vertical velocity, expressed in $\mathrm{m.s^{-1}}$, of the droplet's fall as a function of the fall duration $t$ is given by the function $v$ defined as follows:
For every non-negative real number $t$, $v(t) = 9.81\dfrac{m}{k}\left(1 - \mathrm{e}^{-\frac{k}{m}t}\right)$; the constant $m$ is the mass of the droplet in milligrams and the constant $k$ is a strictly positive coefficient related to air friction.
We recall that instantaneous velocity is the derivative of position. Parts $A$ and $B$ are independent.
Part A - General case

1. Determine the variations of the velocity of the water droplet.
2. Does the droplet slow down during its fall?
3. Show that $\lim_{t \rightarrow +\infty} v(t) = 9.81\dfrac{m}{k}$. This limit is called the terminal velocity of the droplet.
4. A scientist claims that after a fall duration equal to $\dfrac{5m}{k}$, the velocity of the droplet exceeds $99\%$ of its terminal velocity. Is this claim correct?

Part B
In this part, we take $m = 6$ and $k = 3.9$. At a given instant, the instantaneous velocity of this droplet is $15\mathrm{~m.s^{-1}}$.

1. How long ago did the droplet detach from its cloud? Round the answer to the nearest tenth of a second.
2. Deduce the average velocity of this droplet between the moment it detached from the cloud and the instant when its velocity was measured. Round the answer to the nearest tenth of $\mathrm{m.s^{-1}}$.

We consider the function $f$ defined on $\mathbb { R }$ by:
$$f ( x ) = \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + 1 }$$
The representative curve $\mathscr { C }$ of the function $f$ in an orthonormal coordinate system is given.

1. Calculate the limit of the function $f$ at negative infinity and interpret the result graphically.
2. Show that the line with equation $y = 2$ is a horizontal asymptote to the curve $\mathscr { C }$.
3. Calculate $f ^ { \prime } ( x )$, where $f ^ { \prime }$ is the derivative function of $f$, and verify that for all real numbers $x$ we have: $$f ^ { \prime } ( x ) = \frac { f ( x ) } { \mathrm { e } ^ { x } + 1 } .$$
4. Show that the function $f$ is increasing on $\mathbb { R }$.
5. Show that the curve $\mathscr { C }$ passes through the point $\mathrm { I } ( 0 ; 1 )$ and that its tangent at this point has slope 0.5.

Part A

The function $g$ is defined on $[ 0 ; + \infty [$ by $$g ( x ) = 1 - \mathrm { e } ^ { - x } .$$ We admit that the function $g$ is differentiable on $[ 0 ; + \infty [$.

1. Determine the limit of the function $g$ at $+ \infty$.
2. Study the variations of the function $g$ on $[ 0 ; + \infty [$ and draw its variation table.

For each of the following statements, indicate whether it is true or false. You will justify each answer.
Statement 1: For all real numbers $a$ and $b$, $\left( \mathrm{e}^{a+b} \right)^{2} = \mathrm{e}^{2a} + \mathrm{e}^{2b}$.
Statement 2: In the plane with a coordinate system, the tangent line at point A with abscissa 0 to the representative curve of the function $f$ defined on $\mathbb{R}$ by $f(x) = -2 + (3-x)\mathrm{e}^{x}$ has the reduced equation $y = 2x + 1$.
Statement 3: $\lim_{x \rightarrow +\infty} \left( \mathrm{e}^{2x} - \mathrm{e}^{x} + \frac{3}{x} \right) = 0$.
Statement 4: The equation $1 - x + \mathrm{e}^{-x} = 0$ has a unique solution belonging to the interval $[0 ; 2]$.
Statement 5: The function $g$ defined on $\mathbb{R}$ by $g(x) = x^{2} - 5x + \mathrm{e}^{x}$ is convex.

Question 4: Consider the function $f$ defined on $\mathbb{R}$ by $f(x) = 3\mathrm{e}^x - x$.

a. $\lim_{x\rightarrow+\infty} f(x) = 3$	b. $\lim_{x\rightarrow+\infty} f(x) = +\infty$	c. $\lim_{x\rightarrow+\infty} f(x) = -\infty$	\begin{tabular}{l} d. We cannot
determine the limit
of the function $f$
as $x$ tends to
$+\infty$

\hline \end{tabular}

Exercise 1 — Multiple Choice (Exponential function)
For each of the following questions, only one of the four proposed answers is correct.

1. Let $f$ be the function defined on $\mathbb{R}$ by $$f(x) = \frac{x}{\mathrm{e}^{x}}$$ We assume that $f$ is differentiable on $\mathbb{R}$ and we denote $f'$ its derivative function. a. $f'(x) = \mathrm{e}^{-x}$ b. $f'(x) = x\mathrm{e}^{-x}$ c. $f'(x) = (1-x)\mathrm{e}^{-x}$ d. $f'(x) = (1+x)\mathrm{e}^{-x}$
   
2. Let $f$ be a function twice differentiable on the interval $[-3;1]$. The graphical representation of its second derivative function $f''$ is given. We can then affirm that: a. The function $f$ is convex on the interval $[-2;0]$ b. The function $f$ is concave on the interval $[-1;1]$ c. The function $f'$ is decreasing on the interval $[-2;0]$ d. The function $f'$ admits a maximum at $x = -1$
   
3. We consider the function $f$ defined on $\mathbb{R}$ by: $$f(x) = x^3 \mathrm{e}^{-x^2}$$ If $F$ is an antiderivative of $f$ on $\mathbb{R}$, a. $F(x) = -\frac{1}{6}\left(x^3+1\right)\mathrm{e}^{-x^2}$ b. $F(x) = -\frac{1}{4}x^4 \mathrm{e}^{-x^2}$ c. $F(x) = -\frac{1}{2}\left(x^2+1\right)\mathrm{e}^{-x^2}$ d. $F(x) = x^2\left(3-2x^2\right)\mathrm{e}^{-x^2}$
   
4. What is the value of: $$\lim_{x \rightarrow +\infty} \frac{\mathrm{e}^x + 1}{\mathrm{e}^x - 1}$$ a. $-1$ b. $1$ c. $+\infty$ d. does not exist
   
5. We consider the function $f$ defined on $\mathbb{R}$ by $f(x) = \mathrm{e}^{2x+1}$. The only antiderivative $F$ on $\mathbb{R}$ of the function $f$ such that $F(0) = 1$ is the function: a. $x \longmapsto 2\mathrm{e}^{2x+1} - 2\mathrm{e} + 1$ b. $x \longmapsto 2\mathrm{e}^{2x+1} - \mathrm{e}$ c. $x \longmapsto \frac{1}{2}\mathrm{e}^{2x+1} - \frac{1}{2}\mathrm{e} + 1$ d. $x \longmapsto \mathrm{e}^{x^2+x}$
   
6. In a coordinate system, the representative curve of a function $f$ defined and twice differentiable on $[-2;4]$ is drawn. Among the following curves (a, b, c, d), which one represents the function $f''$, the second derivative of $f$?

Consider the function $g$ defined on $\mathbb { R }$ by: $$g ( x ) = \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + 1 } .$$ The representative curve of the function $g$ admits as an asymptote at $+ \infty$ the line with equation: a. $x = 2$; b. $y = 2$; c. $y = 0$; d. $x = - 1$

Exercise 3 — 7 points
Themes: Exponential function and sequence
Part A:
Let $h$ be the function defined on $\mathbb{R}$ by $$h(x) = \mathrm{e}^x - x$$

1. Determine the limits of $h$ at $-\infty$ and $+\infty$.
2. Study the variations of $h$ and draw up its variation table.
3. Deduce that: if $a$ and $b$ are two real numbers such that $0 < a < b$ then $h(a) - h(b) < 0$.

Part B:
Let $f$ be the function defined on $\mathbb{R}$ by $$f(x) = \mathrm{e}^x$$ We denote $\mathscr{C}_f$ its representative curve in a coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$.

1. Determine an equation of the tangent line $T$ to $\mathscr{C}_f$ at the point with abscissa 0.

In the rest of the exercise we are interested in the gap between $T$ and $\mathscr{C}_f$ in the neighbourhood of 0. This gap is defined as the difference of the ordinates of the points of $T$ and $\mathscr{C}_f$ with the same abscissa. We are interested in points with abscissa $\frac{1}{n}$, with $n$ a non-zero natural number. We then consider the sequence $(u_n)$ defined for all non-zero natural numbers $n$ by: $$u_n = \exp\left(\frac{1}{n}\right) - \frac{1}{n} - 1$$

1. Determine the limit of the sequence $(u_n)$.
2. a. Prove that, for all non-zero natural numbers $n$, $$u_{n+1} - u_n = h\left(\frac{1}{n+1}\right) - h\left(\frac{1}{n}\right)$$ where $h$ is the function defined in Part A. b. Deduce the direction of variation of the sequence $(u_n)$.
3. The table below gives approximate values to $10^{-9}$ of the first terms of the sequence $(u_n)$.
   

   $n$	$u_n$
   1	0.718281828
   2	0.148721271
   3	0.062279092
   4	0.034025417
   5	0.021402758
   6	0.014693746
   7	0.010707852
   8	0.008148453
   9	0.006407958
   10	0.005170918

   
   Give the smallest value of the natural number $n$ for which the gap between $T$ and $\mathscr{C}_f$ appears to be less than $10^{-2}$.

Exercise 3 — 6 points

Theme: Exponential function Main topics covered: Sequences; Functions, Exponential function.

Part A

We consider the function $f$ defined for every real $x$ by: $$f ( x ) = 1 + x - \mathrm { e } ^ { 0,5 x - 2 } .$$ We assume that the function $f$ is differentiable on $\mathbb { R }$. We denote $f ^ { \prime }$ its derivative.

1. a. Determine the limit of the function $f$ at $- \infty$. b. Prove that, for every non-zero real $x$, $f ( x ) = 1 + 0,5 x \left( 2 - \frac { \mathrm { e } ^ { 0,5 x } } { 0,5 x } \times \mathrm { e } ^ { - 2 } \right)$. Deduce the limit of the function $f$ at $+ \infty$.
2. a. Determine $f ^ { \prime } ( x )$ for every real $x$. b. Prove that the set of solutions of the inequality $f ^ { \prime } ( x ) < 0$ is the interval $] 4 + 2 \ln ( 2 ) ; + \infty [$.
3. Deduce from the previous questions the variation table of the function $f$ on $\mathbb { R }$. The exact value of the image of $4 + 2 \ln ( 2 )$ by $f$ should be shown.
4. Show that the equation $f ( x ) = 0$ has a unique solution on the interval $[ - 1 ; 0 ]$.

Part B

We consider the sequence $( u _ { n } )$ defined by $u _ { 0 } = 0$ and, for every natural number $n$, $u _ { n + 1 } = f \left( u _ { n } \right)$ where $f$ is the function defined in Part A.

1. a. Prove by induction that, for every natural number $n$, we have: $$u _ { n } \leqslant u _ { n + 1 } \leqslant 4 .$$ b. Deduce that the sequence $( u _ { n } )$ converges. We denote its limit by $\ell$.
2. a. We recall that $f$ satisfies the relation $\ell = f ( \ell )$. Prove that $\ell = 4$. b. We consider the function value written below in the Python language: \begin{verbatim} def valeur (a) : u = 0 n = 0 while u <= a: u=1 + u - exp(0.5*u - 2) n = n+1 return n \end{verbatim} The instruction valeur(3.99) returns the value 12. Interpret this result in the context of the exercise.

For each of the following statements, indicate whether it is true or false. Justify each answer.

1. Statement 1: For all real $x : 1 - \frac { 1 - \mathrm { e } ^ { x } } { 1 + \mathrm { e } ^ { x } } = \frac { 2 } { 1 + \mathrm { e } ^ { - x } }$.
   
2. We consider the function $g$ defined on $\mathbb { R }$ by $g ( x ) = \frac { \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + 1 }$. Statement 2: The equation $g ( x ) = \frac { 1 } { 2 }$ admits a unique solution in $\mathbb { R }$.
   
3. We consider the function $f$ defined on $\mathbb { R }$ by $f ( x ) = x ^ { 2 } \mathrm { e } ^ { - x }$ and we denote $\mathscr { C }$ its curve in an orthonormal coordinate system. Statement 3: The $x$-axis is tangent to the curve $\mathscr { C }$ at only one point.
   
4. We consider the function $h$ defined on $\mathbb { R }$ by $h ( x ) = \mathrm { e } ^ { x } \left( 1 - x ^ { 2 } \right)$. Statement 4: In the plane equipped with an orthonormal coordinate system, the curve representing the function $h$ does not admit an inflection point.
   
5. Statement 5: $\lim _ { x \rightarrow + \infty } \frac { \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + x } = 0$.
   
6. Statement 6: For all real $x , 1 + \mathrm { e } ^ { 2 x } \geqslant 2 \mathrm { e } ^ { x }$.

For every real $x$, the expression $2 + \frac { 3 \mathrm { e } ^ { - x } - 5 } { \mathrm { e } ^ { - x } + 1 }$ is equal to: a. $\frac { 5 - 3 \mathrm { e } ^ { x } } { 1 + \mathrm { e } ^ { x } }$; b. $\frac { 5 + 3 \mathrm { e } ^ { x } } { 1 - \mathrm { e } ^ { x } }$; c. $\frac { 5 + 3 \mathrm { e } ^ { x } } { 1 + \mathrm { e } ^ { x } }$; d. $\frac { 5 - 3 \mathrm { e } ^ { x } } { 1 - \mathrm { e } ^ { x } }$.

Consider the function $g$ defined on $[0; +\infty[$ by $g(t) = \frac{a}{b + \mathrm{e}^{-t}}$ where $a$ and $b$ are two real numbers. We know that $g(0) = 2$ and $\lim_{t \rightarrow +\infty} g(t) = 3$. The values of $a$ and $b$ are:
A. $a = 2$ and $b = 3$
B. $a = 4$ and $b = \frac{4}{3}$
C. $a = 4$ and $b = 1$
D. $a = 6$ and $b = 2$

Exercise 3 — 5 points Theme: exponential function, algorithms For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer receives no points.

1. Statement: The function $f$ defined on $\mathbb{R}$ by $f(x) = \mathrm{e}^x - x$ is convex.
2. Statement: The equation $(2\mathrm{e}^x - 6)(\mathrm{e}^x + 2) = 0$ has $\ln(3)$ as its unique solution in $\mathbb{R}$.
3. Statement: $$\lim_{x \to +\infty} \frac{\mathrm{e}^{2x} - 1}{\mathrm{e}^x - x} = 0.$$
4. Let $f$ be the function defined on $\mathbb{R}$ by $f(x) = (6x + 5)\mathrm{e}^{3x}$ and $F$ the function defined on $\mathbb{R}$ by: $F(x) = (2x + 1)\mathrm{e}^{3x} + 4$. Statement: $F$ is the antiderivative of $f$ on $\mathbb{R}$ that takes the value 5 when $x = 0$.
5. We consider the function \texttt{mystere} defined below which takes a list $L$ of numbers as a parameter. We recall that \texttt{len(L)} represents the length of list $L$. \begin{verbatim} def mystere(L) : S = 0 for i in range(len(L)) : S = S + L[i] return S / len(L) \end{verbatim} Statement: The execution of \texttt{mystere([1, 9, 9, 5, 0, 3, 6, 12, 0, 5])} returns 50.

Question 164
Uma função $f$ é definida por $f(x) = 2^x$. O valor de $f(3) - f(1)$ é
(A) 2 (B) 4 (C) 6 (D) 8 (E) 16

Uma função exponencial é definida por $f(x) = 2^x$. O valor de $f(3) - f(1)$ é
(A) 4 (B) 5 (C) 6 (D) 7 (E) 8

There are electric showers on the market with different power ratings, which represent different consumptions and costs. The power (P) of an electric shower is given by the product between its electrical resistance (R) and the square of the electrical current (i) flowing through it. The consumption of electrical energy (E), in turn, is directly proportional to the power of the device.
Considering the characteristics presented, which of the following graphs represents the relationship between the energy consumed (E) by an electric shower and the electrical current (i) flowing through it?
(A) [graph A]
(B) [graph B]
(C) [graph C]
(D) [graph D]
(E) [graph E]

QUESTION 161
The value of $e^0 + \ln 1$ is
(A) 0
(B) 1
(C) 2
(D) $e$
(E) $e + 1$

QUESTION 173
The function $f(x) = 3^x$ passes through the point
(A) $(0, 0)$
(B) $(0, 1)$
(C) $(1, 0)$
(D) $(1, 3)$
(E) $(3, 1)$

Assume that a type of eucalyptus has an expected exponential growth rate in the first years after planting, modeled by the function $y(t) = a^{t-1}$, in which $y$ represents the height of the plant in meters, $t$ is considered in years, and $a$ is a constant greater than 1. The graph represents the function $y$.
Also assume that $y(0)$ gives the height of the seedling when planted, and it is desired to cut the eucalyptus when the seedlings grow 7.5 m after planting.
The time between planting and cutting, in years, is equal to
(A) 3.
(B) 4.
(C) 6.
(D) $\log_{2} 7$.
(E) $\log_{2} 15$.

The government of a city is concerned about a possible epidemic of an infectious disease caused by bacteria. To decide what measures to take, it must calculate the reproduction rate of the bacteria. In laboratory experiments of a bacterial culture, initially with 40 thousand units, the formula for the population was obtained:
$$p(t) = 40 \cdot 2^{3t}$$
where $t$ is the time, in hours, and $p(t)$ is the population, in thousands of bacteria.
In relation to the initial quantity of bacteria, after 20 min, the population will be
(A) reduced to one third.
(B) reduced to half.
(C) reduced to two thirds.
(D) doubled.
(E) tripled.

A loan was made at a monthly rate of $i\%$, using compound interest, in eight fixed and equal installments of $P$.
The debtor has the possibility of paying off the debt early at any time, paying for this the present value of the remaining installments. After paying the $5^{\text{th}}$ installment, he decides to pay off the debt when paying the $6^{\text{th}}$ installment.
The expression that corresponds to the total amount paid for the loan settlement is
(A) $P \left[ 1 + \frac { 1 } { \left( 1 + \frac { i } { 100 } \right) } + \frac { 1 } { \left( 1 + \frac { i } { 100 } \right) ^ { 2 } } \right]$
(B) $P \left[ 1 + \frac { 1 } { \left( 1 + \frac { i } { 100 } \right) } + \frac { 1 } { \left( 1 + \frac { 2i } { 100 } \right) } \right]$
(C) $P \left[ 1 + \frac { 1 } { \left( 1 + \frac { i } { 100 } \right) ^ { 2 } } + \frac { 1 } { \left( 1 + \frac { i } { 100 } \right) ^ { 2 } } \right]$
(D) $P \left[ \frac { 1 } { \left( 1 + \frac { i } { 100 } \right) } + \frac { 1 } { \left( 1 + \frac { 2i } { 100 } \right) } + \frac { 1 } { \left( 1 + \frac { 3i } { 100 } \right) } \right]$
(E) $P \left[ \frac { 1 } { \left( 1 + \frac { i } { 100 } \right) } + \frac { 1 } { \left( 1 + \frac { i } { 100 } \right) ^ { 2 } } + \frac { 1 } { \left( 1 + \frac { i } { 100 } \right) ^ { 3 } } \right]$

For the function $f ( x ) = \frac { 4 ^ { x } } { 4 ^ { x } + 2 }$, select all correct statements from . [4 points]
ㄱ. $f \left( \frac { 1 } { 2 } \right) = \frac { 1 } { 2 }$ ㄴ. $f ( x ) + f ( 1 - x ) = 1$ ㄷ. $\sum _ { k = 1 } ^ { 100 } f \left( \frac { k } { 101 } \right) = 50$
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄴ
(5) ㄱ, ㄴ, ㄷ