4) Identify this regular polygon, justifying your answer.

\section*{PROBLEM 2}
Let us consider the function $f : \mathbb { R } \rightarrow \mathbb { R }$, periodic with period $T = 4$ whose graph, in the interval $[0; 4]$, is as follows:\\
\includegraphics[max width=\textwidth, alt={}, center]{4247a904-6fcc-4fd2-a50f-9c2a64781d61-3_659_974_1110_536}

As can be seen from Figure 1, the sections $OB, BD, DE$ of the graph are line segments whose endpoints have coordinates: $O(0,0), B(1,1), D(3,-1), E(4,0)$.

1) Establish at which points of its domain the function $f$ is continuous and at which it is differentiable, and verify the existence of the limits: $\lim_{x \rightarrow +\infty} f(x)$ and $\lim_{x \rightarrow +\infty} \frac{f(x)}{x}$; if they exist, determine their value.

Also represent, for $x \in [0; 4]$, the graphs of the functions:

$$\begin{gathered}
g ( x ) = f ^ { \prime } ( x ) \\
h ( x ) = \int _ { 0 } ^ { x } f ( t ) d t
\end{gathered}$$

2) Consider the function:

$$s ( x ) = \sin ( b x )$$

with $b$ a positive real constant; determine $b$ so that $s(x)$ has the same period as $f(x)$.

\section*{Ministry of Education, University and Research}
Prove that the square portion of the plane $OABC$ in Figure 1 is subdivided by the graphs of $f(x)$ and $s(x)$ into 3 distinct parts and determine the probabilities that a point chosen at random inside the square $OABC$ falls in each of the 3 identified parts.

3) Now considering the functions:

$$f ( x ) ^ { 2 } \quad \text { and } \quad s ( x ) ^ { 2 }$$

discuss, also with qualitative arguments, the variations (increase or decrease) of the 3 probability values determined in the previous point.

4) Finally, determine the volume of the solid generated by the rotation around the $y$-axis of the portion of the plane between the graph of the function $h$ for $x \in [0; 3]$ and the $x$-axis.

\section*{QUESTIONNAIRE}
1. Defined the number $E$ as:

$$E = \int _ { 0 } ^ { 1 } x e ^ { x } d x$$

prove that:

$$\int _ { 0 } ^ { 1 } x ^ { 2 } e ^ { x } d x = e - 2 E$$

and express

$$\int _ { 0 } ^ { 1 } x ^ { 3 } e ^ { x } d x$$

in terms of $e$ and $E$.\\
2. A cake in the shape of a cylinder is placed under a plastic dome in the shape of a hemisphere. Prove that the cake occupies less than $3/5$ of the volume of the hemisphere.

3. Knowing that:

$$\lim _ { x \rightarrow 0 } \frac { \sqrt { a x + 2 b } - 6 } { x } = 1$$

determine the values of $a$ and $b$.\\
4. To draw real numbers in the interval $[0,2]$ a random number generator is created that provides numbers distributed, in that interval, with probability density given by the function:

$$f ( x ) = \frac { 3 } { 2 } x ^ { 2 } - \frac { 3 } { 4 } x ^ { 3 }$$

What will be the mean value of the generated numbers?\\
What is the probability that the first number drawn is $4/3$?\\
What is the probability that the second number drawn is less than 1?

\section*{Ministry of Education, University and Research}