There is a quartic function $f ( x )$ with leading coefficient 1, $f ( 0 ) = 3$, and $f ^ { \prime } ( 3 ) < 0$. For a real number $t$, define the set $S$ as $$S = \{ a \mid \text{the function } | f ( x ) - t | \text{ is not differentiable at } x = a \}$$ and let $g ( t )$ be the number of elements in set $S$. When the function $g ( t )$ is discontinuous only at $t = 3$ and $t = 19$, find the value of $f ( - 2 )$. [4 points]

For the function $f ( x ) = k x ^ { 2 } e ^ { - x } ( k > 0 )$ and a real number $t$, let $g ( t )$ be the smaller of the distance from the point $( t , f ( t ) )$ on the curve $y = f ( x )$ to the $x$-axis and the distance to the $y$-axis. What is the maximum value of $k$ such that the function $g ( t )$ is not differentiable at exactly one point? [4 points]
(1) $\frac { 1 } { e }$
(2) $\frac { 1 } { \sqrt { e } }$
(3) $\frac { e } { 2 }$
(4) $\sqrt { e }$
(5) $e$

For a real number $0 < t < 41$, the curve $y = x ^ { 3 } + 2 x ^ { 2 } - 15 x + 5$ and the line $y = t$ intersect at three points. Let the point with the largest $x$-coordinate be $( f ( t ) , t )$ and the point with the smallest $x$-coordinate be $( g ( t ) , t )$. Let $h ( t ) = t \times \{ f ( t ) - g ( t ) \}$. What is the value of $h ^ { \prime } ( 5 )$? [4 points]
(1) $\frac { 79 } { 12 }$
(2) $\frac { 85 } { 12 }$
(3) $\frac { 91 } { 12 }$
(4) $\frac { 97 } { 12 }$
(5) $\frac { 103 } { 12 }$

For two real numbers $a$ and $k$, two functions $f ( x )$ and $g ( x )$ are defined as $$\begin{aligned} & f ( x ) = \begin{cases} 0 & ( x \leq a ) \\ ( x - 1 ) ^ { 2 } ( 2 x + 1 ) & ( x > a ) \end{cases} \\ & g ( x ) = \begin{cases} 0 & ( x \leq k ) \\ 12 ( x - k ) & ( x > k ) \end{cases} \end{aligned}$$ and satisfy the following conditions. (가) The function $f ( x )$ is differentiable on the entire set of real numbers. (나) For all real numbers $x$, $f ( x ) \geq g ( x )$. When the minimum value of $k$ is $\frac { q } { p }$, find the value of $a + p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]

The function $f ( x )$ is a cubic function with leading coefficient 1, and the function $g ( x )$ is a linear function. Define the function $h ( x )$ as $$h ( x ) = \begin{cases} | f ( x ) - g ( x ) | & ( x < 1 ) \\ f ( x ) + g ( x ) & ( x \geq 1 ) \end{cases}$$ If $h ( x )$ is differentiable on the entire set of real numbers, and $h ( 0 ) = 0$, $h ( 2 ) = 5$, find the value of $h ( 4 )$. [4 points]

For a positive number $a$, the function $f(x)$ defined on $x \geq -1$ is $$f(x) = \begin{cases} -x^2 + 6x & (-1 \leq x < 6) \\ a\log_4(x-5) & (x \geq 6) \end{cases}$$ For a real number $t \geq 0$, let $g(t)$ denote the maximum value of $f(x)$ on the closed interval $[t-1, t+1]$. If the minimum value of the function $g(t)$ on the interval $[0, \infty)$ is 5, find the minimum value of the positive number $a$. [4 points]

Consider the function $$f(x) := \frac{1}{3}x^3 \quad \text{if } x \geq 0, \quad f(x) := 0 \quad \text{if } x < 0$$ Justify that $f \in \mathcal{C}^1(\mathbb{R})$ and that $f$ is convex. Give the set of its minimizers.

113- If $f(x) = \text{Max}\{|2x|, |x+1|\}$, then what is the minimum value of $f(x)$?
(1) $\dfrac{1}{3}$ (2) $\dfrac{2}{3}$ (3) $\dfrac{4}{3}$ (4) $2$

121- The set of lengths of inflection points of the curve $y = x|x^2 - 4x|$ is which of the following?
(1) $\left\{\dfrac{4}{3}\right\}$ (2) $\left\{0, \dfrac{4}{3}, 4\right\}$ (3) $\left\{\dfrac{4}{3}, 4\right\}$ (4) $\left\{0, \dfrac{4}{3}\right\}$

10. Given the function:
$$f ( x ) = \left| 4 - x ^ { 2 } \right|$$
verify that it does not satisfy all the hypotheses of Rolle's theorem in the interval $[-3; 3]$ and that nevertheless there exists at least one point in the interval $[-3; 3]$ where the first derivative of $f(x)$ vanishes. Does this example contradict Rolle's theorem? Justify your answer thoroughly.
\footnotetext{Maximum duration of the examination: 6 hours. The use of scientific and/or graphing calculators is permitted provided they are not equipped with symbolic computation capability (O.M. no. 257 Art. 18 paragraph 8). The use of a bilingual dictionary (Italian – language of the country of origin) is permitted for candidates whose native language is not Italian. It is not permitted to leave the Institute before 3 hours have elapsed from the dictation of the theme. }

17. The function $f ( x ) = ( x 2 - 1 ) | x 2 - 3 x + 2 |$ is NOT differentiable at:
(A) - 1
(B) 0
(C) 1
(D) 2

29. The function $f ( x ) = \int - 1 x t$ (et-1)(t-1)(t-2)3(t-3)5dt has a local minimum at $x =$
(A) 0
(B) 1
(C) 2
(D) 3

28. Let $f ( x ) = \left\{ \begin{array} { c l l } | x | & \text { for } 0 < | x | \leq 2 \\ 1 & \text { for } & x = 0 . \end{array} \right.$
Then at $\mathrm { x } = 0 , \mathrm { f }$ has :
(A) A local maximum
(B) no local maximum
(C) a local minimum
(D) no extremum

The total number of local maxima and local minima of the function $$f ( x ) = \begin{cases} ( 2 + x ) ^ { 3 } , & - 3 < x \leq - 1 \\ x ^ { 2 / 3 } , & - 1 < x < 2 \end{cases}$$ is
(A) 0
(B) 1
(C) 2
(D) 3

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ be respectively given by $f(x) = |x| + 1$ and $g(x) = x^2 + 1$. Define $h : \mathbb{R} \rightarrow \mathbb{R}$ by $$h(x) = \begin{cases} \max\{f(x), g(x)\} & \text{if } x \leq 0 \\ \min\{f(x), g(x)\} & \text{if } x > 0 \end{cases}$$ The number of points at which $h(x)$ is not differentiable is

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be given by $$f ( x ) = \left\{ \begin{aligned} x ^ { 5 } + 5 x ^ { 4 } + 10 x ^ { 3 } + 10 x ^ { 2 } + 3 x + 1 , & x < 0 \\ x ^ { 2 } - x + 1 , & 0 \leq x < 1 \\ \frac { 2 } { 3 } x ^ { 3 } - 4 x ^ { 2 } + 7 x - \frac { 8 } { 3 } , & 1 \leq x < 3 \\ ( x - 2 ) \log _ { e } ( x - 2 ) - x + \frac { 10 } { 3 } , & x \geq 3 \end{aligned} \right.$$ Then which of the following options is/are correct?
(A) $f$ is increasing on $( - \infty , 0 )$
(B) $f ^ { \prime }$ has a local maximum at $x = 1$
(C) $f$ is onto
(D) $f ^ { \prime }$ is NOT differentiable at $x = 1$

Let $f ( x ) = \left\{ \begin{array} { c c } x ^ { 3 } - x ^ { 2 } + 10 x - 7 , & x \leq 1 \\ - 2 x + \log _ { 2 } \left( b ^ { 2 } - 4 \right) , & x > 1 \end{array} \right.$ Then the set of all values of $b$, for which $f ( x )$ has maximum value at $x = 1$, is:
(1) $( - 6 , - 2 )$
(2) $( 2,6 )$
(3) $[ - 6 , - 2 ) \cup ( 2,6 ]$
(4) $[ - \sqrt { 6 } , - 2 ) \cup ( 2 , \sqrt { 6 } ]$

Let a function $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined as: $f ( x ) = \begin{cases} \int _ { 0 } ^ { x } ( 5 - | t - 3 | ) d t , & x > 4 \\ x ^ { 2 } + b x , & x \leq 4 \end{cases}$ where $b \in \mathbb { R }$. If $f$ is continuous at $x = 4$, then which of the following statements is NOT true?
(1) $f$ is not differentiable at $x = 4$
(2) $f ^ { \prime } ( 3 ) + f ^ { \prime } ( 5 ) = \frac { 35 } { 4 }$
(3) $f$ is increasing in $\left( - \infty , \frac { 1 } { 8 } \right) \cup ( 8 , \infty )$
(4) $f$ has a local minima at $x = \frac { 1 } { 8 }$

The sum of all local minimum values of the function
$$f ( x ) = \left\{ \begin{array} { l r } 1 - 2 x , & x < - 1 \\ \frac { 1 } { 3 } ( 7 + 2 | x | ) , & - 1 \leq x \leq 2 \\ \frac { 11 } { 18 } ( x - 4 ) ( x - 5 ) , & x > 2 \end{array} \right.$$
is
(1) $\frac { 157 } { 72 }$
(2) $\frac { 131 } { 72 }$
(3) $\frac { 171 } { 72 }$
(4) $\frac { 167 } { 72 }$