EXERCISE - B

Main topics covered: Function study, exponential function; Differential equations

Part I

Let us consider the differential equation $$y' = -0.4y + 0.4$$ where $y$ denotes a function of the variable $t$, defined and differentiable on $[0; +\infty[$.

1. a. Determine a particular constant solution of this differential equation. b. Deduce the set of solutions of this differential equation. c. Determine the function $g$, solution of this differential equation, which satisfies $g(0) = 10$.

Part II

Let $p$ be the function defined and differentiable on the interval $[0; +\infty[$ by $$p(t) = \frac{1}{g(t)} = \frac{1}{1 + 9\mathrm{e}^{-0.4t}}$$

1. Determine the limit of $p$ at $+\infty$.
2. Show that $p'(t) = \frac{3.6\mathrm{e}^{-0.4t}}{\left(1 + 9\mathrm{e}^{-0.4t}\right)^{2}}$ for all $t \in [0; +\infty[$.
3. a. Show that the equation $p(t) = \frac{1}{2}$ has a unique solution $\alpha$ on $[0; +\infty[$. b. Determine an approximate value of $\alpha$ to $10^{-1}$ near using a calculator.

Part III

1. $p$ denotes the function from Part II. Verify that $p$ is a solution of the differential equation $y' = 0.4y(1 - y)$ with the initial condition $y(0) = \frac{1}{10}$ where $y$ denotes a function defined and differentiable on $[0; +\infty[$.
2. In a developing country, in the year 2020, 10\% of schools have access to the internet.
   A voluntary equipment policy is implemented and we are interested in the evolution of the proportion of schools with access to the internet. We denote $t$ the time elapsed, expressed in years, since the year 2020. The proportion of schools with access to the internet at time $t$ is modelled by $p(t)$. Interpret in this context the limit from question II 1 then the approximate value of $\alpha$ from question II 3. b. as well as the value $p(0)$.

Part A: Let $g$ be the function defined on $\mathbb{R}$ by: $$g(x) = 2\mathrm{e}^{\frac{-1}{3}x} + \frac{2}{3}x - 2$$

1. We admit that the function $g$ is differentiable on $\mathbb{R}$ and we denote $g^{\prime}$ its derivative function. Show that, for every real number $x$: $$g^{\prime}(x) = \frac{-2}{3}e^{-\frac{1}{3}x} + \frac{2}{3}.$$
2. Deduce the direction of variation of the function $g$ on $\mathbb{R}$.
3. Determine the sign of $g(x)$, for every real $x$.

Part B:

1. Consider the differential equation $$(E):\quad 3y^{\prime} + y = 0.$$ Solve the differential equation (E).
2. Determine the particular solution whose representative curve, in a coordinate system of the plane, passes through the point $\mathrm{M}(0;2)$.
3. Let $f$ be the function defined on $\mathbb{R}$ by: $$f(x) = 2\mathrm{e}^{-\frac{1}{3}x}$$ and $\mathscr{C}_f$ its representative curve. a. Show that the tangent line $(\Delta_0)$ to the curve $\mathscr{C}_f$ at the point $\mathrm{M}(0;2)$ has an equation of the form: $$y = -\frac{2}{3}x + 2$$ b. Study, on $\mathbb{R}$, the position of this curve $\mathscr{C}_f$ relative to the tangent line $(\Delta_0)$.

Part C:

1. Let A be the point on the curve $\mathscr{C}_f$ with abscissa $a$, where $a$ is any real number. Show that the tangent line $(\Delta_a)$ to the curve $\mathscr{C}_f$ at point A intersects the $x$-axis at a point P with abscissa $a+3$.
2. Explain the construction of the tangent line $(\Delta_{-2})$ to the curve $\mathscr{C}_f$ at point B with abscissa $-2$.

EXERCISE B - Differential equation
Part A: Determination of a function $f$ and resolution of a differential equation
Consider the function $f$ defined on $\mathbb{R}$ by: $$f(x) = \mathrm{e}^x + ax + b\mathrm{e}^{-x}$$ where $a$ and $b$ are real numbers that we propose to determine in this part. In the plane with a coordinate system with origin O, the curve $\mathscr{C}$, representing the function $f$, and the tangent line $(T)$ to the curve $\mathscr{C}$ at the point with abscissa 0 are shown.

1. By reading the graph, give the values of $f(0)$ and $f'(0)$.
2. Using the expression of the function $f$, express $f(0)$ as a function of $b$ and deduce the value of $b$.
3. We admit that the function $f$ is differentiable on $\mathbb{R}$ and we denote by $f'$ its derivative function. a. Give, for every real $x$, the expression of $f'(x)$. b. Express $f'(0)$ as a function of $a$. c. Using the previous questions, determine $a$, then deduce the expression of $f(x)$.
4. Consider the differential equation: $$( E ) : \quad y' + y = 2\mathrm{e}^x - x - 1$$ a. Verify that the function $g$ defined on $\mathbb{R}$ by: $$g(x) = \mathrm{e}^x - x + 2\mathrm{e}^{-x}$$ is a solution of equation $(E)$. b. Solve the differential equation $y' + y = 0$. c. Deduce all solutions of equation $(E)$.

Part B: Study of the function $g$ on $[1;+\infty[$

1. Verify that for every real $x$, we have: $$\mathrm{e}^{2x} - \mathrm{e}^x - 2 = (\mathrm{e}^x - 2)(\mathrm{e}^x + 1)$$
2. Deduce a factored expression of $g'(x)$, for every real $x$.
3. We will admit that, for all $x \in [1;+\infty[$, $\mathrm{e}^x - 2 > 0$. Study the direction of variation of the function $g$ on $[1;+\infty[$.

Main topics covered: Differential equations; exponential function.
We consider the differential equation
$$\text { (E) } y ^ { \prime } = y + 2 x \mathrm { e } ^ { x }$$
We seek the set of functions defined and differentiable on the set $\mathbb { R }$ of real numbers that are solutions to this equation.

1. Let $u$ be the function defined on $\mathbb { R }$ by $u ( x ) = x ^ { 2 } \mathrm { e } ^ { x }$. We admit that $u$ is differentiable and we denote $u ^ { \prime }$ its derivative function. Prove that $u$ is a particular solution of $( E )$.
2. Let $f$ be a function defined and differentiable on $\mathbb { R }$. We denote $g$ the function defined on $\mathbb { R }$ by: $$g ( x ) = f ( x ) - u ( x )$$ a. Prove that if the function $f$ is a solution of the differential equation $( E )$ then the function $g$ is a solution of the differential equation: $y ^ { \prime } = y$. We admit that the converse of this property is also true. b. Using the solution of the differential equation $y ^ { \prime } = y$, solve the differential equation (E).
3. Study of the function $u$ a. Study the sign of $u ^ { \prime } ( x )$ for $x$ varying in $\mathbb { R }$. b. Draw the table of variations of the function $u$ on $\mathbb { R }$ (limits are not required). c. Determine the largest interval on which the function $u$ is concave.

EXERCISE A Main topics covered: Exponential function, convexity, differentiation, differential equations
This exercise consists of three independent parts. Below is represented, in an orthonormal coordinate system, a portion of the representative curve $\mathscr { C }$ of a function $f$ defined on $\mathbb { R }$.
Consider the points $\mathrm { A } ( 0 ; 2 )$ and $\mathrm { B } ( 2 ; 0 )$.

Part 1

Knowing that the curve $\mathscr { C }$ passes through A and that the line (AB) is tangent to the curve $\mathscr { C }$ at point A, give by reading the graph:

1. The value of $f ( 0 )$ and that of $f ^ { \prime } ( 0 )$.
2. An interval on which the function $f$ appears to be convex.

Part 2

We denote $(E)$ the differential equation $$y ^ { \prime } = - y + \mathrm { e } ^ { - x }$$ It is admitted that $g : x \longmapsto x \mathrm { e } ^ { - x }$ is a particular solution of $(E)$.

1. Give all solutions on $\mathbb { R }$ of the differential equation $( H ) : y ^ { \prime } = - y$.
2. Deduce all solutions on $\mathbb { R }$ of the differential equation $(E)$.
3. Knowing that the function $f$ is the particular solution of $(E)$ which satisfies $f ( 0 ) = 2$, determine an expression of $f ( x )$ as a function of $x$.

Part 3

It is admitted that for every real number $x , f ( x ) = ( x + 2 ) \mathrm { e } ^ { - x }$.

1. Recall that $f ^ { \prime }$ denotes the derivative function of the function $f$. a. Show that for all $x \in \mathbb { R } , f ^ { \prime } ( x ) = ( - x - 1 ) \mathrm { e } ^ { - x }$. b. Study the sign of $f ^ { \prime } ( x )$ for all $x \in \mathbb { R }$ and draw up the table of variations of $f$ on $\mathbb { R }$. Neither the limit of $f$ at $- \infty$ nor the limit of $f$ at $+ \infty$ will be specified. Calculate the exact value of the extremum of $f$ on $\mathbb { R }$.
2. Recall that $f ^ { \prime \prime }$ denotes the second derivative function of the function $f$. a. Calculate for all $x \in \mathbb { R } , f ^ { \prime \prime } ( x )$. b. Can we assert that $f$ is convex on the interval $[ 0 ; + \infty [$?

Exercise 1

PART A Consider the differential equation $$( E ) : \quad y ^ { \prime } + \frac { 1 } { 4 } y = 20 \mathrm { e } ^ { - \frac { 1 } { 4 } x } ,$$ with unknown $y$, a function defined and differentiable on the interval $[ 0 ; + \infty [$.

1. Determine the value of the real number $a$ such that the function $g$ defined on the interval $[ 0 ; + \infty [$ by $g ( x ) = a x \mathrm { e } ^ { - \frac { 1 } { 4 } x }$ is a particular solution of the differential equation $( E )$.
2. Consider the differential equation $$\left( E ^ { \prime } \right) : \quad y ^ { \prime } + \frac { 1 } { 4 } y = 0 ,$$ with unknown $y$, a function defined and differentiable on the interval $[ 0 ; + \infty [$. Determine the solutions of the differential equation ( $E ^ { \prime }$ ).
3. Deduce the solutions of the differential equation ( $E$ ).
4. Determine the solution $f$ of the differential equation ( $E$ ) such that $f ( 0 ) = 8$.

PART B Consider the function $f$ defined on the interval $[ 0 ; + \infty [$ by $$f ( x ) = ( 20 x + 8 ) \mathrm { e } ^ { - \frac { 1 } { 4 } x }$$ It is admitted that the function $f$ is differentiable on the interval $\left[ 0 ; + \infty \left[ \right. \right.$ and we denote $f ^ { \prime }$ its derivative function on the interval $\left[ 0 ; + \infty \left[ \right. \right.$. Moreover, it is admitted that $\lim _ { x \rightarrow + \infty } f ( x ) = 0$.

1. a. Justify that, for every positive real number $x$, $$f ^ { \prime } ( x ) = ( 18 - 5 x ) \mathrm { e } ^ { - \frac { 1 } { 4 } x }$$ b. Deduce the table of variations of the function $f$. The exact value of the maximum of the function $f$ on the interval $[ 0 ; + \infty [$ will be specified.
2. In this question we are interested in the equation $f ( x ) = 8$. a. Justify that the equation $f ( x ) = 8$ admits a unique solution, denoted $\alpha$, in the interval [14; 15]. b. Copy and complete the table below by running step by step the solution\_equation function opposite, written in Python language
   

   $a$	14
   $b$	15
   $b - a$	1
   $m$	14,5
   \begin{tabular}{ l } Condition
   $f ( m ) > 8$

   & FALSE & & & & \hline \end{tabular}
   \begin{verbatim} from math import exp def f(x) : return (20* x +8)*exp(-1/4* x) def solution_equation() : a,b = 14,15 while b-a>0.1: m = (a+b)/2 if f (m) > 8 : |a=m else : | b=m return a,b \end{verbatim}
   c. What is the objective of the solution\_equation function in the context of the question?

Consider the differential equation $$\left( E_0 \right) : \quad y^{\prime} = y$$ where $y$ is a differentiable function of the real variable $x$.

1. Prove that the unique constant function solution of the differential equation $\left( E_0 \right)$ is the zero function.
2. Determine all solutions of the differential equation $(E_0)$.

Consider the differential equation $$(E) : \quad y^{\prime} = y - \cos(x) - 3\sin(x)$$ where $y$ is a differentiable function of the real variable $x$.

1. The function $h$ is defined on $\mathbb{R}$ by $h(x) = 2\cos(x) + \sin(x)$. It is admitted that it is differentiable on $\mathbb{R}$. Prove that the function $h$ is a solution of the differential equation $(E)$.
2. Consider a function $f$ defined and differentiable on $\mathbb{R}$. Prove that: ``$f$ is a solution of $(E)$'' is equivalent to ``$f - h$ is a solution of $\left(E_0\right)$''.
3. Deduce all solutions of the differential equation $(E)$.
4. Determine the unique solution $g$ of the differential equation $(E)$ such that $g(0) = 0$.
5. Calculate: $$\int_{0}^{\frac{\pi}{2}} \left[ -2\mathrm{e}^{x} + \sin(x) + 2\cos(x) \right] \mathrm{d}x$$

This exercise is a multiple choice questionnaire (MCQ) comprising five questions. The five questions are independent. For each question, only one of the four answers is correct.

1. The solution $f$ of the differential equation $y' = -3y + 7$ such that $f(0) = 1$ is the function defined on $\mathbb{R}$ by:
   A. $f(x) = \mathrm{e}^{-3x}$
   B. $f(x) = -\frac{4}{3}\mathrm{e}^{-3x} + \frac{7}{3}$
   C. $f(x) = \mathrm{e}^{-3x} + \frac{7}{3}$
   D. $f(x) = -\frac{10}{3}\mathrm{e}^{-3x} - \frac{7}{3}$
   
2. The curve of a function $f$ defined on $[0; +\infty[$ is given below.
   A bound for the integral $I = \int_{1}^{5} f(x)\,\mathrm{d}x$ is:
   A. $0 \leqslant I \leqslant 4$
   B. $1 \leqslant I \leqslant 5$
   C. $5 \leqslant I \leqslant 10$
   D. $10 \leqslant I \leqslant 15$
   
3. Consider the function $g$ defined on $\mathbb{R}$ by $g(x) = x^2 \ln\left(x^2 + 4\right)$.
   Then $\int_{0}^{2} g'(x)\,\mathrm{d}x$ equals, to $10^{-1}$ near:
   A. 4.9
   B. 8.3
   C. 1.7
   D. 7.5
   
4. A teacher teaches the mathematics specialization in a class of 31 final year students. She wants to form a group of 5 students. In how many different ways can she form such a group of 5 students?
   A. $31^5$
   B. $31 \times 30 \times 29 \times 28 \times 27$
   C. $31 + 30 + 29 + 28 + 27$
   D. $\binom{31}{5}$
   
5. The teacher is now interested in the other specialization of the 31 students in her group:
   • 10 students chose the physics-chemistry specialization;
   • 20 students chose the SES specialization;
   • 1 student chose the Spanish LLCE specialization.
   She wants to form a group of 5 students containing exactly 3 students who chose the SES specialization. In how many different ways can she form such a group?
   A. $\binom{20}{3} \times \binom{11}{2}$
   B. $\binom{20}{3} + \binom{11}{2}$
   C. $\binom{20}{3}$
   D. $20^3 \times 11^2$

By noting that $\int_{-\infty}^{+\infty} \frac{\partial f}{\partial t}(t, x) \exp(-\mathrm{i} 2\pi \xi x) \mathrm{d}x = \mathcal{F}\left(\frac{\partial f}{\partial t}(t, \cdot)\right)(\xi)$ and using question 7, show that, for any real $\xi$ and any real $t > 0$, $\frac{\partial \hat{f}}{\partial t}(t, \xi) = -4\pi^{2} \xi^{2} \hat{f}(t, \xi)$.

Show that, for all $\xi \in \mathbb{R}$, there exists a real $K(\xi)$ such that for all $t \in \mathbb{R}_{+}^{*}$, $\hat{f}(t, \xi) = K(\xi) \exp\left(-4\pi^{2} \xi^{2} t\right)$.

Give, depending on the sign of $\lambda$, the form of harmonic functions with separable variables.

Determine the radial harmonic functions of $\mathbb{R}^2 \setminus \{(0,0)\}$, that is, the functions $f$ belonging to $\mathcal{H}\left(\mathbb{R}^2 \setminus \{(0,0)\}\right)$ such that $(r,\theta) \mapsto f(r\cos(\theta), r\sin(\theta))$ is independent of $\theta$.

Let $a, b, r_1$ and $r_2$ be four real numbers such that $0 < r_1 < r_2$. Determine a function $f$ of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$ such that $$\begin{cases} \Delta f = 0 \\ f(x,y) = a & \text{if } \|(x,y)\| = r_1 \\ f(x,y) = b & \text{if } \|(x,y)\| = r_2 \end{cases}$$

We assume here that $\lambda = 0$. What are the $2\pi$-periodic solutions of the differential equation (II.2): $$z''(\theta) + \lambda z(\theta) = 0$$

We assume here that $\lambda = 0$. Solve the differential equation (II.1) $$r^2 z''(r) + r z'(r) - \lambda z(r) = 0$$ on $\mathbb{R}^{+*}$.

We seek to determine the non-zero harmonic functions on $\mathbb{R}^2$ with separable variables, that is, functions $f$ that can be written in the form $f(x,y) = u(x)v(y)$. We are given two functions $u$ and $v$, of class $\mathcal{C}^2$ on $\mathbb{R}$, not identically zero, and we set $$\forall (x,y) \in \mathbb{R}^2, \quad f(x,y) = u(x)v(y)$$ We assume that $f$ is harmonic on $\mathbb{R}^2$.
Give, depending on the sign of $\lambda$, the form of harmonic functions with separable variables.

Let $f$ be a real function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$. We set, for all $(r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R}$, $$g(r,\theta) = f(r\cos(\theta), r\sin(\theta))$$ Determine the radial harmonic functions of $\mathbb{R}^2 \setminus \{(0,0)\}$, that is, the functions $f$ belonging to $\mathcal{H}(\mathbb{R}^2 \setminus \{(0,0)\})$ such that $(r,\theta) \mapsto f(r\cos(\theta), r\sin(\theta))$ is independent of $\theta$.

Let $a, b, r_1$ and $r_2$ be four real numbers such that $0 < r_1 < r_2$. Determine a function $f$ of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$ such that $$\begin{cases} \Delta f = 0 \\ f(x,y) = a & \text{if } \|(x,y)\| = r_1 \\ f(x,y) = b & \text{if } \|(x,y)\| = r_2 \end{cases}$$

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ We assume here that $\lambda = 0$. What are the $2\pi$-periodic solutions of (II.2): $$z''(\theta) + \lambda z(\theta) = 0$$

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ We assume here that $\lambda = 0$. Solve (II.1) on $\mathbb{R}^{+*}$: $$r^2 z''(r) + r z'(r) - \lambda z(r) = 0$$

Let $\alpha$ be a real number. We denote $f_{\alpha} : x \longmapsto (1-x)^{-\alpha}$.
Specify the domain of definition $D$ of $f_{\alpha}$. Justify that $f_{\alpha}$ is of class $C^{1}$ on $D$ and give a first-order linear differential equation satisfied by $f_{\alpha}$ on $D$.

Let $\alpha$ be a real number. We denote $f_{\alpha} : x \longmapsto (1-x)^{-\alpha}$.
State Cauchy's theorem for a scalar first-order linear differential equation and prove that, for all $x \in ]-1,1[$, $$f_{\alpha}(x) = \sum_{n=0}^{+\infty} L_{n}(\alpha) \frac{x^{n}}{n!}.$$

Solve the differential equation $xy' - y = 0$ on each of the intervals $]0, R[$ and $]-R, 0[$ then on the interval $]-R, R[$.

Mathematics Specialty - EXERCISE II (20 points)

The questions in Part I can be treated independently. In this exercise, $K$ and $a$ are strictly positive real constants.

Part I - Preliminary Studies

Consider the differential equation $\left( E _ { 1 } \right) : z ^ { \prime } ( t ) + z ( t ) = \frac { 1 } { K }$, where $z$ is a function defined and differentiable on $[ 0 ; + \infty [$. II-1- Give the general solution of $( E _ { 1 } )$ on the interval $[ 0 ; + \infty [$. Consider the function $f$ defined for every positive real $t$ by: $f ( t ) = \frac { 10 } { 1 + a e ^ { - t } }$. II-2- Complete the table of variations of $f$ on the interval $[ 0 ; + \infty [$. Specify the value of $f$ at $0$ and the limit of $f$ at $+ \infty$. II-3- Determine, as a function of $a$, the set of solutions of the equation $f ( t ) = 5$.

Part II - Evolution of a Marmot Population

Let $y _ { 0 }$ be a strictly positive real number. We study the evolution of a marmot population, which initially numbers $y _ { 0 }$ thousand individuals. We admit that the size of the population, expressed in thousands of individuals, after $t$ years (with $t \geq 0$) is a function $y$ differentiable on $[ 0 ; + \infty [$, solution of the differential equation: $$\left( E _ { 2 } \right) : y ^ { \prime } ( t ) = y ( t ) \left( 1 - \frac { y ( t ) } { K } \right)$$ The constant $K$ is called the carrying capacity of the environment, expressed in thousands of individuals. We admit that there exists a unique function $y$ solution of $\left( E _ { 2 } \right)$ that satisfies $y ( 0 ) = y _ { 0 }$. We admit that this function takes strictly positive values on $[ 0 ; + \infty [$. We set $z ( t ) = \frac { 1 } { y ( t ) }$ for every positive real $t$. II-4-a- Express $z ^ { \prime } ( t )$ as a function of $y ^ { \prime } ( t )$ and $y ( t )$. II-4-b- We wish to show that $z$ is a solution of $\left( E _ { 1 } \right)$ if, and only if, $y$ is a solution of $\left( E _ { 2 } \right)$. Complete:

• Line 1 using an expression involving $z ^ { \prime } ( t )$ and $z ( t )$;
• Line 2 and Line 3 using an expression involving $y ^ { \prime } ( t )$ and $y ( t )$.

II-5-a- Deduce the general solution of $( E _ { 2 } )$. II-5-b- We admit that the unique solution $y$ of $\left( E _ { 2 } \right)$ satisfying $y ( 0 ) = y _ { 0 }$ is written in the form $y ( t ) = \frac { K } { 1 + a e ^ { - t } }$. Express $a$ as a function of $y _ { 0 }$ and $K$. In a certain valley with carrying capacity $K = 10$, the marmots have disappeared. Scientists wish to reintroduce $y _ { 0 }$ thousand marmots, with $0 < y _ { 0 } < 10$. In the remainder of the exercise, we will take $K = 10$. II-6- Justify that the value of $a$ obtained in question II-5-b- is indeed strictly positive. II-7-a- Using the result from question II-3-, give the value of $a$ such that $y ( 5 ) = 5$. II-7-b- Deduce the exact value of $y _ { 0 }$ such that $y ( 5 ) = 5$. Justify your answer. II-7-c- The calculator gives $0.0669285092$ as the result of the calculation of the value of $y _ { 0 }$ from the previous question. What is the minimum number of marmots to reintroduce so that at least $5$ thousand marmots are present after $5$ years following their reintroduction?

The curve satisfying the differential equation, $y d x - \left( x + 3 y ^ { 2 } \right) d y = 0$ and passing through the point $( 1,1 )$ also passes through the point
(1) $\left( \frac { 1 } { 4 } , - \frac { 1 } { 2 } \right)$
(2) $\left( - \frac { 1 } { 3 } , \frac { 1 } { 3 } \right)$
(3) $\left( \frac { 1 } { 4 } , \frac { 1 } { 2 } \right)$
(4) $\left( \frac { 1 } { 3 } , - \frac { 1 } { 3 } \right)$