4. Provide a justified estimate of the number of tiles that, having a stain in the non-coloured part, will be damaged at the end of the production cycle.
Ministry of Education, University and Research
PROBLEM 2
Let us consider the function $f_k : \mathbb{R} \rightarrow \mathbb{R}$ defined as: $$f_k(x) = -x^3 + kx + 9$$ with $k \in \mathbb{Z}$. 1. Let $\Gamma_k$ be the graph of the function, verify that for any value of the parameter $k$ the line $r_k$, tangent to $\Gamma_k$ at the point with abscissa 0, and the line $s_k$, tangent to $\Gamma_k$ at the point with abscissa 1, meet at a point $M$ with abscissa $\frac{2}{3}$. 2. After verifying that $k = 1$ is the maximum positive integer for which the ordinate of point $M$ is less than 10, study the behaviour of the function $f_1(x)$, determining its stationary and inflection points and sketching its graph. 3. Let $T$ be the triangle bounded by the lines $r_1$, $s_1$ and the $x$-axis, determine the probability that, taking at random a point $P(x_p, y_p)$ inside $T$, it lies above $\Gamma_1$ (that is, that $y_p > f_1(x)$ for such point $P$). 4. In the figure a point $N \in \Gamma_1$ and a portion of the graph $\Gamma_1$ are highlighted. The normal line to $\Gamma_1$ at $N$ (that is, the perpendicular to the tangent line to $\Gamma_1$ at that point) passes through the origin of the axes $O$. The graph $\Gamma_1$ has three points with this property. Prove, more generally, that the graph of any polynomial of degree $n > 0$ cannot have more than $2n - 1$ points at which the normal line to the graph passes through the origin. [Figure]
Ministry of Education, University and Research
QUESTIONNAIRE
1. Prove that the volume of a cylinder inscribed in a cone is less than half the volume of the cone. 2. There are two identical unbalanced dice in the shape of a regular tetrahedron with faces numbered from 1 to 4. When rolling each of the two dice, the probability of getting 1 is twice the probability of getting 2, which in turn is twice the probability of getting 3, which in turn is twice the probability of getting 4. If the two dice are rolled simultaneously, what is the probability that two equal numbers come out? 3. Determine the values of $k$ such that the line with equation $y = -4x + k$ is tangent to the curve with equation $y = x^3 - 4x^2 + 5$. 4. Considering the function $f(x) = \frac{3x - e^{\sin x}}{5 + e^{-x} - \cos x}$, determine, if they exist, the values of $\lim_{x \rightarrow +\infty} f(x)$, $\lim_{x \rightarrow -\infty} f(x)$, justifying the answers provided adequately.
4. Provide a justified estimate of the number of tiles that, having a stain in the non-coloured part, will be damaged at the end of the production cycle.
\section*{Ministry of Education, University and Research}
\section*{PROBLEM 2}
Let us consider the function $f_k : \mathbb{R} \rightarrow \mathbb{R}$ defined as:
$$f_k(x) = -x^3 + kx + 9$$
with $k \in \mathbb{Z}$.
1. Let $\Gamma_k$ be the graph of the function, verify that for any value of the parameter $k$ the line $r_k$, tangent to $\Gamma_k$ at the point with abscissa 0, and the line $s_k$, tangent to $\Gamma_k$ at the point with abscissa 1, meet at a point $M$ with abscissa $\frac{2}{3}$.
2. After verifying that $k = 1$ is the maximum positive integer for which the ordinate of point $M$ is less than 10, study the behaviour of the function $f_1(x)$, determining its stationary and inflection points and sketching its graph.
3. Let $T$ be the triangle bounded by the lines $r_1$, $s_1$ and the $x$-axis, determine the probability that, taking at random a point $P(x_p, y_p)$ inside $T$, it lies above $\Gamma_1$ (that is, that $y_p > f_1(x)$ for such point $P$).
4. In the figure a point $N \in \Gamma_1$ and a portion of the graph $\Gamma_1$ are highlighted. The normal line to $\Gamma_1$ at $N$ (that is, the perpendicular to the tangent line to $\Gamma_1$ at that point) passes through the origin of the axes $O$. The graph $\Gamma_1$ has three points with this property. Prove, more generally, that the graph of any polynomial of degree $n > 0$ cannot have more than $2n - 1$ points at which the normal line to the graph passes through the origin.\\
\includegraphics[max width=\textwidth, alt={}, center]{9c1becc2-d505-467f-bee3-3bef03778b0a-3_527_1125_1852_463}
\section*{Ministry of Education, University and Research}
\section*{QUESTIONNAIRE}
1. Prove that the volume of a cylinder inscribed in a cone is less than half the volume of the cone.
2. There are two identical unbalanced dice in the shape of a regular tetrahedron with faces numbered from 1 to 4. When rolling each of the two dice, the probability of getting 1 is twice the probability of getting 2, which in turn is twice the probability of getting 3, which in turn is twice the probability of getting 4. If the two dice are rolled simultaneously, what is the probability that two equal numbers come out?
3. Determine the values of $k$ such that the line with equation $y = -4x + k$ is tangent to the curve with equation $y = x^3 - 4x^2 + 5$.\\
4. Considering the function $f(x) = \frac{3x - e^{\sin x}}{5 + e^{-x} - \cos x}$, determine, if they exist, the values of $\lim_{x \rightarrow +\infty} f(x)$, $\lim_{x \rightarrow -\infty} f(x)$, justifying the answers provided adequately.