$$f ( x ) = \begin{cases} \frac { 2 } { x } & \text { for } x < - 1 \\ x ^ { 2 } - 3 & \text { for } - 1 \leq x \leq 2 \\ 4 x - 3 & \text { for } x > 2 \end{cases}$$
Let $f$ be the function defined above. At what values of $x$, if any, is $f$ not differentiable?
(A) $x = - 1$ only
(B) $x = 2$ only
(C) $x = - 1$ and $x = - 2$
(D) $f$ is differentiable for all values of $x$.

11. If $x + 7 y = 29$ is an equation of the line normal to the graph of $f$ at the point $( 1,4 )$, then $f ^ { \prime } ( 1 ) =$
(A) 7
(B) $\frac { 1 } { 7 }$
(C) $- \frac { 1 } { 7 }$
(D) $- \frac { 7 } { 29 }$
(E) - 7

6. Consider the curve defined by $2 y ^ { 3 } + 6 x ^ { 2 } y - 12 x ^ { 2 } + 6 y = 1$.
(a) Show that $\frac { d y } { d x } = \frac { 4 x - 2 x y } { x ^ { 2 } + y ^ { 2 } + 1 }$.
(b) Write an equation of each horizontal tangent line to the curve.
(c) The line through the origin with slope - 1 is tangent to the curve at point $P$. Find the $x$ - and $y$-coordinates of point $P$.

18. An equation of the line tangent to the graph of $y = x + \cos x$ at the point $( 0,1 )$ is
(A) $y = 2 x + 1$
(B) $y = x + 1$
(C) $y = x$
(D) $y = x - 1$
(E) $y = 0$

Let $f$ be the real-valued function defined by $f ( x ) = \sqrt { 1 + 6 x }$. (a) Give the domain and range of $f$. (b) Determine the slope of the line tangent to the graph of $f$ at $x = 4$. (c) Determine the y -intercept of the line tangent to the graph of f at $\mathrm { x } = 4$. (d) Give the coordinates of the point on the graph of $f$ where the tangent line is parallel to $y = x + 12$.

In the figure above, line $\ell$ is tangent to the graph of $y = \frac { 1 } { x ^ { 2 } }$ at point $P$, with coordinates $\left( w , \frac { 1 } { w ^ { 2 } } \right)$, where $w > 0$. Point $Q$ has coordinates $( w , 0 )$. Line $\ell$ crosses the $x$-axis at point $R$, with coordinates $( k , 0 )$.
(a) Find the value of $k$ when $w = 3$.
(b) For all $w > 0$, find $k$ in terms of $w$.
(c) Suppose that $w$ is increasing at the constant rate of 7 units per second. When $w = 5$, what is the rate of change of $k$ with respect to time?
(d) Suppose that $w$ is increasing at the constant rate of 7 units per second. When $w = 5$, what is the rate of change of the area of $\triangle PQR$ with respect to time? Determine whether the area is increasing or decreasing at this instant.

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 $\lim _ { x \rightarrow 0 ^ { + } } f ( x )$.

A metal wire of length 8 centimeters (cm) is heated at one end. The table above gives selected values of the temperature $T ( x )$, in degrees Celsius ( ${ } ^ { \circ } \mathrm { C } $ ), of the wire $x \mathrm {~cm}$ from the heated end. The function $T$ is decreasing and twice differentiable.
(a) Estimate $T ^ { \prime } ( 7 )$. Show the work that leads to your answer. Indicate units of measure.
(b) Write an integral expression in terms of $T ( x )$ for the average temperature of the wire. Estimate the average temperature of the wire using a trapezoidal sum with the four subintervals indicated by the data in the table. Indicate units of measure.
(c) Find $\int _ { 0 } ^ { 8 } T ^ { \prime } ( x ) d x$, and indicate units of measure. Explain the meaning of $\int _ { 0 } ^ { 8 } T ^ { \prime } ( x ) d x$ in terms of the temperature of the wire.
(d) Are the data in the table consistent with the assertion that $T ^ { \prime \prime } ( x ) > 0$ for every $x$ in the interval $0 < x < 8$ ? Explain your answer.

The temperature of water in a tub at time $t$ is modeled by a strictly increasing, twice-differentiable function $W$, where $W ( t )$ is measured in degrees Fahrenheit and $t$ is measured in minutes. At time $t = 0$, the temperature of the water is $55 ^ { \circ } \mathrm { F }$. The water is heated for 30 minutes, beginning at time $t = 0$. Values of $W ( t )$ at selected times $t$ for the first 20 minutes are given in the table above.
(a) Use the data in the table to estimate $W ^ { \prime } ( 12 )$. Show the computations that lead to your answer. Using correct units, interpret the meaning of your answer in the context of this problem.
(b) Use the data in the table to evaluate $\int _ { 0 } ^ { 20 } W ^ { \prime } ( t ) d t$. Using correct units, interpret the meaning of $\int _ { 0 } ^ { 20 } W ^ { \prime } ( t ) d t$ in the context of this problem.
(c) For $0 \leq t \leq 20$, the average temperature of the water in the tub is $\frac { 1 } { 20 } \int _ { 0 } ^ { 20 } W ( t ) d t$. Use a left Riemann sum with the four subintervals indicated by the data in the table to approximate $\frac { 1 } { 20 } \int _ { 0 } ^ { 20 } W ( t ) d t$. Does this approximation overestimate or underestimate the average temperature of the water over these 20 minutes? Explain your reasoning.
(d) For $20 \leq t \leq 25$, the function $W$ that models the water temperature has first derivative given by $W ^ { \prime } ( t ) = 0.4 \sqrt { t } \cos ( 0.06 t )$. Based on the model, what is the temperature of the water at time $t = 25$ ?

The function $f$ is defined by $f ( x ) = \frac { x } { x + 2 }$. What points $( x , y )$ on the graph of $f$ have the property that the line tangent to $f$ at $( x , y )$ has slope $\frac { 1 } { 2 }$ ?
(A) $( 0,0 )$ only
(B) $\left( \frac { 1 } { 2 } , \frac { 1 } { 5 } \right)$ only
(C) $( 0,0 )$ and $( - 4,2 )$
(D) $( 0,0 )$ and $\left( 4 , \frac { 2 } { 3 } \right)$
(E) There are no such points.

Let $f$ be the function defined by $f ( x ) = e ^ { x } \cos x$.
(a) Find the average rate of change of $f$ on the interval $0 \leq x \leq \pi$.
(b) What is the slope of the line tangent to the graph of $f$ at $x = \frac { 3 \pi } { 2 }$ ?
(c) Find the absolute minimum value of $f$ on the interval $0 \leq x \leq 2 \pi$. Justify your answer.
(d) Let $g$ be a differentiable function such that $g \left( \frac { \pi } { 2 } \right) = 0$. The graph of $g ^ { \prime }$, the derivative of $g$, is shown below. Find the value of $\lim _ { x \rightarrow \pi / 2 } \frac { f ( x ) } { g ( x ) }$ or state that it does not exist. Justify your answer.

Consider the curve defined by the equation $x^2 + 3y + 2y^2 = 48$. It can be shown that $\frac{dy}{dx} = \frac{-2x}{3 + 4y}$.
(a) There is a point on the curve near $(2, 4)$ with $x$-coordinate 3. Use the line tangent to the curve at $(2, 4)$ to approximate the $y$-coordinate of this point.
(b) Is the horizontal line $y = 1$ tangent to the curve? Give a reason for your answer.
(c) The curve intersects the positive $x$-axis at the point $(\sqrt{48}, 0)$. Is the line tangent to the curve at this point vertical? Give a reason for your answer.
(d) For time $t \geq 0$, a particle is moving along another curve defined by the equation $y^3 + 2xy = 24$. At the instant the particle is at the point $(4, 2)$, the $y$-coordinate of the particle's position is decreasing at a rate of 2 units per second. At that instant, what is the rate of change of the $x$-coordinate of the particle's position with respect to time?

The function $f$ is twice differentiable for $x > 0$ with $f ( 1 ) = 15$ and $f ^ { \prime \prime } ( 1 ) = 20$. Values of $f ^ { \prime }$, the derivative of $f$, are given for selected values of $x$ in the table above.

$x$	1	1.1	1.2	1.3	1.4
$f ^ { \prime } ( x )$	8	10	12	13	14.5

(a) Write an equation for the line tangent to the graph of $f$ at $x = 1$. Use this line to approximate $f ( 1.4 )$.
(b) Use a midpoint Riemann sum with two subintervals of equal length and values from the table to approximate $\int _ { 1 } ^ { 1.4 } f ^ { \prime } ( x ) d x$. Use the approximation for $\int _ { 1 } ^ { 1.4 } f ^ { \prime } ( x ) d x$ to estimate the value of $f ( 1.4 )$. Show the computations that lead to your answer.
(c) Use Euler's method, starting at $x = 1$ with two steps of equal size, to approximate $f ( 1.4 )$. Show the computations that lead to your answer.
(d) Write the second-degree Taylor polynomial for $f$ about $x = 1$. Use the Taylor polynomial to approximate $f ( 1.4 )$.

The function $f$ is defined by $f ( x ) = \frac { x } { x + 2 }$. What points $( x , y )$ on the graph of $f$ have the property that the line tangent to $f$ at $( x , y )$ has slope $\frac { 1 } { 2 }$ ?
(A) $( 0,0 )$ only
(B) $\left( \frac { 1 } { 2 } , \frac { 1 } { 5 } \right)$ only
(C) $( 0,0 )$ and $( - 4,2 )$
(D) $( 0,0 )$ and $\left( 4 , \frac { 2 } { 3 } \right)$
(E) There are no such points.

Let $f$ be the function defined by $f(x) = \frac{3}{2x^2 - 7x + 5}$.
(a) Find the slope of the line tangent to the graph of $f$ at $x = 3$.
(b) Find the $x$-coordinate of each critical point of $f$ in the interval $1 < x < 2.5$. Classify each critical point as the location of a relative minimum, a relative maximum, or neither. Justify your answers.
(c) Using the identity that $\frac{3}{2x^2 - 7x + 5} = \frac{2}{2x - 5} - \frac{1}{x - 1}$, evaluate $\int_{5}^{\infty} f(x)\, dx$ or show that the integral diverges.
(d) Determine whether the series $\sum_{n=5}^{\infty} \frac{3}{2n^2 - 7n + 5}$ converges or diverges. State the conditions of the test used for determining convergence or divergence.

In the plane with an orthonormal coordinate system ($\mathrm { O } ; \vec { \imath } , \vec { \jmath }$), the representative curve $\mathscr { C }$ of a function $f$ defined and differentiable on the interval $] 0 ; + \infty [$ is given.
We have the following information:

• the points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ have coordinates $(1,0)$, $(1,2)$, $(0,2)$ respectively;
• the curve $\mathscr { C }$ passes through point B and the line (BC) is tangent to $\mathscr { C }$ at B;
• there exist two positive real numbers $a$ and $b$ such that for every strictly positive real number $x$, $$f ( x ) = \frac { a + b \ln x } { x } .$$

1. 1. [a.] Using the graph, give the values of $f ( 1 )$ and $f ^ { \prime } ( 1 )$.
   2. [b.] Verify that for every strictly positive real number $x , f ^ { \prime } ( x ) = \frac { ( b - a ) - b \ln x } { x ^ { 2 } }$.
   3. [c.] Deduce the real numbers $a$ and $b$.
2. 1. [a.] Justify that for every real number $x$ in the interval $] 0 , + \infty [$, $f ^ { \prime } ( x )$ has the same sign as $- \ln x$.
   2. [b.] Determine the limits of $f$ at 0 and at $+ \infty$. We may note that for every strictly positive real number $x$, $f ( x ) = \frac { 2 } { x } + 2 \frac { \ln x } { x }$.
   3. [c.] Deduce the table of variations of the function $f$.
3. 1. [a.] Prove that the equation $f ( x ) = 1$ has a unique solution $\alpha$ on the interval $] 0,1 ]$.
   2. [b.] By similar reasoning, we prove that there exists a unique real number $\beta$ in the interval $] 1 , + \infty [$ such that $f ( \beta ) = 1$. Determine the integer $n$ such that $n < \beta < n + 1$.
4. The following algorithm is given.
   

   	\begin{tabular}{l} Variables:
   $a , b$ and $m$ are real numbers.
   Initialization:
   Assign to $a$ the value 0.
   Assign to $b$ the value 1.
   Processing:
   While $b - a > 0.1$
   Assign to $m$ the value $\frac { 1 } { 2 } ( a + b )$.
   If $f ( m ) < 1$ then Assign to $a$ the value $m$. Otherwise Assign to $b$ the value $m$.
   End If.
   End While.
   Output:
   Display $a$.
   Display $b$.

   \hline \end{tabular}
   
   1. [a.] Run this algorithm by completing the table below, which you will copy onto your answer sheet.
      

      	step 1	step 2	step 3	step 4	step 5
      $a$	0
      $b$	1
      $b - a$
      $m$

      
      
   2. [b.] What do the values displayed by this algorithm represent?
   3. [c.] Modify the algorithm above so that it displays the two bounds of an interval containing $\beta$ with amplitude $10 ^ { - 1 }$.
5. The purpose of this question is to prove that the curve $\mathscr { C }$ divides the rectangle OABC into two regions of equal area.
   1. [a.] Justify that this amounts to proving that $\int _ { \frac { 1 } { \mathrm { e } } } ^ { 1 } f ( x ) \mathrm { d } x = 1$.
   2. [b.] By noting that the expression of $f ( x )$ can be written as $\frac { 2 } { x } + 2 \times \frac { 1 } { x } \times \ln x$, complete the proof.

We consider the curve $\mathscr{C}$ with equation $y = \mathrm{e}^{x}$.
For every strictly positive real $m$, we denote by $\mathscr{D}_{m}$ the line with equation $y = mx$.

1. In this question, we choose $m = \mathrm{e}$.
   Prove that the line $\mathscr{D}_{\mathrm{e}}$, with equation $y = \mathrm{e}x$, is tangent to the curve $\mathscr{C}$ at its point with abscissa 1.
2. Conjecture, according to the values taken by the strictly positive real $m$, the number of intersection points of the curve $\mathscr{C}$ and the line $\mathscr{D}_{m}$.
3. Prove this conjecture.

Exercise 1 (5 points)
The plane is equipped with an orthogonal coordinate system $(\mathrm{O}, \mathrm{I}, \mathrm{J})$.
We denote by $\Gamma$ the representative curve of the function $g$ defined on the interval $]0; 1]$ by $g(x) = \ln x$. Let $a$ be a real number in the interval $]0; 1]$. We denote by $M_a$ the point on the curve $\Gamma$ with abscissa $a$ and $d_a$ the tangent line to the curve $\Gamma$ at the point $M_a$. This line $d_a$ intersects the $x$-axis at point $N_a$ and the $y$-axis at point $P_a$. We are interested in the area of triangle $\mathrm{O}N_aP_a$ as the real number $a$ varies in the interval $]0; 1]$.
In this question, we study the particular case where $a = 0.2$.
a. Determine graphically an estimate of the area of triangle $\mathrm{O}N_{0.2}P_{0.2}$ in square units.
b. Determine an equation of the tangent line $d_{0.2}$.
c. Calculate the exact value of the area of triangle $\mathrm{O}N_{0.2}P_{0.2}$.

EXERCISE-A

Main topics covered: convexity, logarithm function

Part I: graphical readings

$f$ denotes a function defined and differentiable on $\mathbb{R}$. We give below the representative curve of the derivative function $f'$.

With the precision allowed by the graph, answer the following questions

1. Determine the slope of the tangent line to the curve of function $f$ at 0.
2. a. Give the variations of the derivative function $f'$. b. Deduce an interval on which $f$ is convex.

Part II: function study

The function $f$ is defined on $\mathbb{R}$ by $$f(x) = \ln\left(x^{2} + x + \frac{5}{2}\right)$$

1. Calculate the limits of function $f$ at $+\infty$ and at $-\infty$.
2. Determine an expression $f'(x)$ of the derivative function of $f$ for all $x \in \mathbb{R}$.
3. Deduce the table of variations of $f$. Be sure to place the limits in this table.
4. a. Justify that the equation $f(x) = 2$ has a unique solution $\alpha$ in the interval $\left[-\frac{1}{2}; +\infty\right[$. b. Give an approximate value of $\alpha$ to $10^{-1}$ near.
5. The function $f'$ is differentiable on $\mathbb{R}$. We admit that, for all $x \in \mathbb{R}$, $f''(x) = \frac{-2x^{2} - 2x + 4}{\left(x^{2} + x + \frac{5}{2}\right)^{2}}$. Determine the number of inflection points of the representative curve of $f$.

Question 1: Consider the function $g$ defined on $]0;+\infty[$ by $g(x) = x^2 + 2x - \frac{3}{x}$. An equation of the tangent line to the curve representing $g$ at the point with abscissa 1 is:

a. $y = 7(x-1)$

b. $y = x-1$

c. $y = 7x+7$

d. $y = x+1$

The graph opposite gives the graphical representation $\mathscr { C } _ { f }$ in an orthogonal coordinate system of a function $f$ defined and differentiable on $\mathbb { R }$. We denote $f ^ { \prime }$ the derivative function of $f$. We are given points A with coordinates (0; 5) and B with coordinates (1; 20). Point C is the point on the curve $\mathscr { C } _ { f }$ with abscissa $-2.5$. The line (AB) is tangent to the curve $\mathscr { C } _ { f }$ at point A. Questions 1 to 3 relate to this same function $f$.
We can assert that: a. $f ^ { \prime } ( - 0.5 ) = 0$ b. if $x \in ] - \infty ; - 0.5 \left[ \right.$, then $f ^ { \prime } ( x ) < 0$ c. $f ^ { \prime } ( 0 ) = 15$ d. the derivative function $f ^ { \prime }$ does not change sign on $\mathbb { R }$.

The graph opposite gives the graphical representation $\mathscr { C } _ { f }$ in an orthogonal coordinate system of a function $f$ defined and differentiable on $\mathbb { R }$. We denote $f ^ { \prime }$ the derivative function of $f$. We are given points A with coordinates (0; 5) and B with coordinates (1; 20). Point C is the point on the curve $\mathscr { C } _ { f }$ with abscissa $-2.5$. The line (AB) is tangent to the curve $\mathscr { C } _ { f }$ at point A. Questions 1 to 3 relate to this same function $f$.
We admit that the function $f$ represented above is defined on $\mathbb { R }$ by $f ( x ) = ( a x + b ) \mathrm { e } ^ { x }$, where $a$ and $b$ are two real numbers and that its curve intersects the x-axis at the point with coordinates ($-0.5$; 0). We can assert that: a. $a = 10$ and $b = 5$ b. $a = 2.5$ and $b = -0.5$ c. $a = -1.5$ and $b = 5$ d. $a = 0$ and $b = 5$

The graph opposite gives the graphical representation $\mathscr { C } _ { f }$ in an orthogonal coordinate system of a function $f$ defined and differentiable on $\mathbb { R }$. We denote $f ^ { \prime }$ the derivative function of $f$. We are given points A with coordinates (0; 5) and B with coordinates (1; 20). Point C is the point on the curve $\mathscr { C } _ { f }$ with abscissa $-2.5$. The line (AB) is tangent to the curve $\mathscr { C } _ { f }$ at point A. Questions 1 to 3 relate to this same function $f$.
We admit that the second derivative of the function $f$ is defined on $\mathbb { R }$ by: $$f ^ { \prime \prime } ( x ) = ( 10 x + 25 ) \mathrm { e } ^ { x }$$ We can assert that: a. The function $f$ is convex on $\mathbb { R }$ b. The function $f$ is concave on $\mathbb { R }$ c. Point C is the unique inflection point of $\mathscr { C } _ { f }$ d. $\mathscr { C } _ { f }$ has no inflection point

Let $f$ be a function defined and differentiable on $\mathbb { R }$. We consider the points $\mathrm { A } ( 1 ; 3 )$ and $\mathrm { B } ( 3 ; 5 )$. We give below $\mathscr { C } _ { f }$ the representative curve of $f$ in an orthogonal coordinate system of the plane, as well as the tangent line (AB) to the curve $\mathscr { C } _ { f }$ at point A.
The three parts of the exercise can be worked on independently.

Part A

1. Determine graphically the values of $f ( 1 )$ and $f ^ { \prime } ( 1 )$.
2. The function $f$ is defined by the expression $f ( x ) = \ln \left( a x ^ { 2 } + 1 \right) + b$, where $a$ and $b$ are positive real numbers. a. Determine the expression of $f ^ { \prime } ( x )$. b. Determine the values of $a$ and $b$ using the previous results.

Part B

It is admitted that the function $f$ is defined on $\mathbb { R }$ by $$f ( x ) = \ln \left( x ^ { 2 } + 1 \right) + 3 - \ln ( 2 )$$

1. Show that $f$ is an even function.
2. Determine the limits of $f$ at $+ \infty$ and at $- \infty$.
3. Determine the expression of $f ^ { \prime } ( x )$. Study the direction of variation of the function $f$ on $\mathbb { R }$. Draw up the table of variations of $f$ showing the exact value of the minimum as well as the limits of $f$ at $- \infty$ and $+ \infty$.
4. Using the table of variations of $f$, give the values of the real number $k$ for which the equation $f ( x ) = k$ admits two solutions.
5. Solve the equation $f ( x ) = 3 + \ln 2$.

Part C

We recall that the function $f$ is defined on $\mathbb{R}$ by $f ( x ) = \ln \left( x ^ { 2 } + 1 \right) + 3 - \ln ( 2 )$.

1. Conjecture, by graphical reading, the abscissas of any inflection points of the curve $\mathscr { C } _ { f }$.
2. Show that, for any real number $x$, we have: $f ^ { \prime \prime } ( x ) = \frac { 2 \left( 1 - x ^ { 2 } \right) } { \left( x ^ { 2 } + 1 \right) ^ { 2 } }$.
3. Deduce the largest interval on which the function $f$ is convex.

Consider the function $f$ defined on $\mathbb { R }$ by :
$$f ( x ) = \frac { 1 } { 1 + \mathrm { e } ^ { - 3 x } }$$
We denote $\mathscr { C } _ { f }$ its representative curve in an orthogonal coordinate system of the plane. We name A the point with coordinates $\left( 0 ; \frac { 1 } { 2 } \right)$ and B the point with coordinates $\left( 1 ; \frac { 5 } { 4 } \right)$. Below we have drawn the curve $\mathscr { C } _ { f }$ and $\mathscr { T }$ the tangent line to the curve $\mathscr { C } _ { f }$ at the point with abscissa 0.

Part A: graphical readings

In this part, results will be obtained by graphical reading. No justification is required.

1. Determine the reduced equation of the tangent line $\mathscr { T }$.
2. Give the intervals on which the function $f$ appears to be convex or concave.

Part B : study of the function

1. We admit that the function $f$ is differentiable on $\mathbb { R }$.

Determine the expression of its derivative function $f ^ { \prime }$.
2. Justify that the function $f$ is strictly increasing on $\mathbb { R }$.
3. a. Determine the limit at $+ \infty$ of the function $f$. b. Determine the limit at $- \infty$ of the function $f$.
4. Determine the exact value of the solution $\alpha$ of the equation $f ( x ) = 0,99$.

Part C : Tangent line and convexity

1. Determine by calculation an equation of the tangent line $\mathscr { T }$ to the curve $\mathscr { C } _ { f }$ at the point with abscissa 0.

We admit that the function $f$ is twice differentiable on $\mathbb { R }$. We denote $f ^ { \prime \prime }$ the second derivative function of the function $f$. We admit that $f ^ { \prime \prime }$ is defined on $\mathbb { R }$ by:
$$f ^ { \prime \prime } ( x ) = \frac { 9 \mathrm { e } ^ { - 3 x } \left( \mathrm { e } ^ { - 3 x } - 1 \right) } { \left( 1 + \mathrm { e } ^ { - 3 x } \right) ^ { 3 } } .$$

1. Study the sign of the function $f ^ { \prime \prime }$ on $\mathbb { R }$.
2. a. Indicate, by justifying, on which interval(s) the function $f$ is convex. b. What does point A represent for the curve $\mathscr { C } _ { f }$ ? c. Deduce the relative position of the tangent line $\mathscr { T }$ and the curve $\mathscr { C } _ { f }$. Justify the answer.