If $x = 2$ is an extremum point of the function $f(x) = (x-1)(x-2)(x-a)$, then $f(0) = $ \_\_\_\_

For each value of $a$, the graph of $f _ { a }$ has exactly two extrema. Determine the value of $a$ for which the graph of the function $f _ { a }$ has an extremum at the point $x = 3$.
Given is the function $f : x \mapsto 2 \cdot \left( ( \ln x ) ^ { 2 } - 1 \right)$ defined on $\mathbb { R } ^ { + }$. Figure 1 shows the graph $G _ { f }$ of $f$.
[Figure]
Fig. 1
(1a) [5 marks] Show that $x = e ^ { - 1 }$ and $x = e$ are the only zeros of $f$, and calculate the coordinates of the minimum point $T$ of $G _ { f }$. (for verification: $f ^ { \prime } ( x ) = \frac { 4 } { x } \cdot \ln x$ )
(1b) [6 marks] Show that $G _ { f }$ has exactly one inflection point $W$, and determine its coordinates and the equation of the tangent to $G _ { f }$ at the point $W$. (for verification: $x$-coordinate of $W : e$ )
(1c) [6 marks] Justify that $\lim _ { x \rightarrow 0 } f ^ { \prime } ( x ) = - \infty$ and $\lim _ { x \rightarrow + \infty } f ^ { \prime } ( x ) = 0$ hold. Give $f ^ { \prime } ( 0,5 )$ and $f ^ { \prime } ( 10 )$ to one decimal place and draw the graph of the derivative function $f ^ { \prime }$ taking into account all previous results in Figure 1.
(1d) [3 marks] Justify using Figure 1 that there are two values $c \in ] 0 ; 6 ]$ for which $\int _ { e ^ { - 1 } } ^ { c } f ( x ) \mathrm { dx } = 0$ holds.
The rational function $h : x \mapsto 1,5 x - 4,5 + \frac { 1 } { x }$ with $x \in \mathbb { R } \backslash \{ 0 \}$ provides a good approximation for $f$ in a certain range.
(1e) [2 marks] Specify the equations of the two asymptotes of the graph of $h$.
(1f) [5 marks] In the fourth quadrant, $G _ { f }$ together with the $x$-axis and the lines with equations $x = 1$ and $x = 2$ enclose a region whose area is approximately 1.623. Determine the percentage deviation from this value if the function $h$ is used as an approximation for the function $f$ when calculating the area.
By reflecting $G _ { f }$ across the line $x = 4$, the graph of a function $g$ defined on $] - \infty ; 8 [$ is created. This graph is denoted by $G _ { g }$. (2a) [2 marks] Draw $G _ { g }$ in Figure 1.
(2b) [3 marks] The described reflection of $G _ { f }$ across the line $x = 4$ can be replaced by a reflection of $G _ { f }$ across the $y$-axis followed by a translation. Describe this translation and specify $a , b \in \mathbb { R }$ such that $g ( x ) = f ( a x + b )$ for $x \in ] - \infty ; 8 [$.
In the following, the ``w-shaped'' curve $k$ is considered, which consists of the part of $G _ { f }$ restricted to $0,2 \leq x \leq 4$ and the part of $G _ { g }$ restricted to $4 < x \leq 7,8$. The curve $k$ is translated 12 units in the negative $z$-direction. The area swept out in this process serves as a model for a 12-meter-long aquarium, which is completed by two flat walls at the front and back to form a basin (see Figure 2). Here, one unit of length in the coordinate system corresponds to one meter in reality.
[Figure]
Fig. 2
(2c) [3 marks] The aquarium walls form a tunnel at the bottom through which visitors can walk. Calculate the size of the angle that the left and right tunnel walls enclose with each other.
The aquarium is completely filled with water.
(2d) [2 marks] Calculate the maximum water depth of the aquarium.
\footnotetext{(c) \href{http://Abiturloesung.de}{Abiturloesung.de} }
(2e) [3 marks] The volume of water in the aquarium can be calculated analogously to the volume of a prism with base area $G$ and height $h$. Explain that the term $24 \cdot \int _ { 0,2 } ^ { 4 } ( f ( 0,2 ) - f ( x ) ) \mathrm { dx }$ describes the water volume in the completely filled aquarium in cubic meters.

(1) [5 marks] Given is the function $f : x \mapsto \frac { e ^ { 2 x } } { x }$ with domain $D _ { f } = \mathbb { R } \backslash \{ 0 \}$. Determine the location and type of the extremum point of the graph of $f$.
Given is the function $f : x \mapsto 1 - \frac { 1 } { x ^ { 2 } }$ defined in $\mathbb { R } \backslash \{ 0 \}$, which has zeros $x _ { 1 } = - 1$ and $x _ { 2 } = 1$. Figure 1 shows the graph of $f$, which is symmetric with respect to the y-axis. Furthermore, the line $g$ with equation $y = - 3$ is given. [Figure]
(2a) [1 marks] Show that one of the points where $g$ intersects the graph of $f$ has the x-coordinate $\frac { 1 } { 2 }$.
(2b) [4 marks] Determine by calculation the area enclosed by the graph of $f$, the x-axis, and the line $g$.
The adjacent Figure 2 shows the graph of a function $f$. [Figure]
(3a) [3 marks] One of the following graphs I, II, and III belongs to the first derivative function of $f$. Specify this graph. Justify why the other two graphs are not suitable. [Figure] [Figure] [Figure]
(3b) [2 marks] The function $F$ is an antiderivative of $f$. Specify the monotonicity behavior of $F$ on the interval $[ 1 ; 3 ]$. Justify your statement.
)$} Consider a family of functions $h _ { k }$ with $k \in \mathbb { R } ^ { + }$, which differ only in their respective domains $D _ { k }$. It holds that $h _ { k } : x \mapsto \cos x$ with $D _ { k } = [ 0 ; k ]$. Figure 4 shows the graph of the function $h _ { 7 }$. Specify the largest possible value of $k$ such that the corresponding function $h _ { k }$ is invertible. For this value of $k$, sketch the graph of the inverse function of $h _ { k }$ in Figure 4 and pay particular attention to the intersection point of the graphs of the function and its inverse. [Figure]
(4b) [2 marks] Specify the term of a function $j$ defined on $\mathbb { R }$ and invertible that satisfies the following condition: The graph of $j$ and the graph of the inverse function of $j$ have no common point.
Given is the function $f : x \mapsto 2 - \ln ( x - 1 )$ with maximal domain $D _ { f }$. The graph of $f$ is denoted by $G _ { f }$.

State the coordinates of the minimum point of the graph of the function $h$ defined in $\mathbb { R }$ with $h ( x ) = - g ( x - 3 )$.
Subtask Part A 4b $( 3 \mathrm { marks } )$ The graph of an antiderivative of $g$ passes through $P$. Sketch this graph in Figure 2.
Given is the function $f : x \mapsto 2 e ^ { - \frac { 1 } { 8 } x ^ { 2 } }$ defined in $\mathbb { R }$. Figure 3 shows the graph $G _ { f }$ of $f$, which has the x-axis as a horizontal asymptote.
[Figure]
Fig. 3
(1a) [2 marks] Calculate the coordinates of the intersection point of $G _ { f }$ with the y-axis and prove by calculation that $G _ { f }$ is symmetric with respect to the y-axis.
(1b) [5 marks] The point $W \left( - 2 \left\lvert \, 2 e ^ { - \frac { 1 } { 2 } } \right. \right)$ is one of the two inflection points of $G _ { f }$. The tangent to $G _ { f }$ at point $W$ is denoted by $w$. Determine an equation of $w$ and calculate the point where $w$ intersects the x-axis. (for verification: $f ^ { \prime } ( x ) = - \frac { 1 } { 2 } x \cdot e ^ { - \frac { 1 } { 8 } x ^ { 2 } }$ )
For each value $c \in \mathbb { R } ^ { + }$, consider the rectangle with vertices $P ( - c \mid 0 ) , Q ( c \mid 0 )$, $R ( c \mid f ( c ) )$ and $S$.
(1c) [1 marks] Draw the rectangle PQRS in Figure 3 for $c = 2$.
(1d) [3 marks] Calculate the value of $c$ for which $\overline { \mathrm { QR } } = 1$ holds.
(1e) [3 marks] State the side lengths of rectangle PQRS as a function of $c$ and justify that the area of the rectangle is given by the term $A ( c ) = 4 c \cdot e ^ { - \frac { 1 } { 8 } c ^ { 2 } }$.
(1f) [4 marks] There is a value of $c$ for which the area $A ( c )$ of rectangle PQRS is maximal. Calculate this value of $c$.
For $k \in \mathbb { R }$, consider the functions $f _ { k } : x \mapsto f ( x ) + k$ defined in $] - \infty ; 0 ]$. Thus $f _ { 0 } ( x ) = f ( x )$, where $f _ { 0 }$ and $f$ differ in their domain.
(1g) [4 marks] Justify using the first derivative of $f _ { k }$ that $f _ { k }$ is invertible for every value of $k$. Sketch the graph of the inverse function of $f _ { 0 }$ in Figure 3.
(1h) [2 marks] State all values of $k$ for which the graph of $f _ { k }$ and the graph of the inverse function of $f _ { k }$ have no common point.
[Figure]
Fig. 4
Figure 4 shows a house with a roof dormer, whose front is shown schematically in Figure 5. The front is described by a model as the region enclosed by the graph $G _ { f }$ of the function $f$ from Part B Subtask 1, the x-axis, and the lines with equations $x = - 4$ and $x = 4$. Here, one unit of length in the coordinate system corresponds to one meter in reality.
[Figure]
Fig. 5
(2a) [2 marks] State the width and height of the front of the roof dormer.
In the front of the roof dormer there is a window. In the model, the window corresponds to the region enclosed by the graph of the function $g$ with $g ( x ) = a x ^ { 4 } + b$ and suitable values $a , b \in \mathbb { R }$ with the x-axis (see Figure 5).
(2b) [2 marks] Justify that $a$ is negative and $b$ is positive.
To determine the area of the front of the roof dormer, an antiderivative $F$ of $f$ is considered.
(2c) [2 marks] One of the graphs I, II and III is the graph of $F$. Justify that this is Graph I by giving one reason each for why Graph II and Graph III do not apply. [Figure] [Figure] [Figure]
(2d) [5 marks] Now determine the area of the entire front of the roof dormer (including the window) using the graph of $F$ from Part B Subtask 2c. Describe, incorporating this area, the essential steps of a solution method by which the value of $a$ could be calculated so that with a window height of 1.50 m, the part of the front of the roof dormer shown shaded in Figure 5 has an area of $6 \mathrm {~m} ^ { 2 }$.
(2e) [5 marks] In order to calculate an approximate value for the length of the upper profile line of the front of the roof dormer, $G _ { f }$ in the range $- 4 \leq x \leq 4$ is approximated by four circular arcs that transition seamlessly into one another and are congruent to each other. One of these circular arcs extends in the range $0 \leq x \leq 2$ and is part of the circle with center $M ( 0 \mid - 1 )$ and radius 3. Calculate the central angle of the circular sector corresponding to this circular arc and use it to determine the desired approximate value.

Given is the function $h$ with $h ( x ) = x ^ { 2 } \cdot \mathrm { e } ^ { - x } , x \in \mathbb { R }$.
(1) (i) Show: $h ^ { \prime } ( x ) = \mathrm { e } ^ { - x } \cdot \left( - x ^ { 2 } + 2 x \right)$.
(ii) Calculate the coordinates and the type of local extreme points of the graph of $h$.
[For verification: The local maximum point of $h$ is $x = 2$.]
(2) The points $P ( 0 \mid 0 ) , Q ( r \mid 0 )$ and $R ( r \mid h ( r ) )$ form the vertices of a triangle $P Q R$ for $0 \leq r \leq 10$. Determine $r$ so that the area of triangle $P Q R$ is maximal.
(3) Describe how the graph of $j$ with $j ( x ) = 3 \cdot ( x - 2 ) ^ { 2 } \cdot \mathrm { e } ^ { - ( x - 2 ) }$ is obtained from the graph of $h$. Give the local maximum point of the graph of the function $j$.

(1) Determine the coordinates of the inflection point of the graph of $f _ { k }$ as a function of $k$ by calculation. [For verification: For the $x$-coordinate $x _ { w }$ of the inflection point, we have: $x _ { w } = k$.] The inflection points of the graphs of $f _ { k }$ with $k \geq - 0,5$ lie on the graph of the function $w$ with $w ( x ) = \mathrm { e } ^ { - x } \cdot x ^ { 2 } , x \geq - 0,5$. The graph of $w$ is called the locus curve of the inflection points of the function family. (2) (i) Show: $w ^ { \prime } ( x ) = \mathrm { e } ^ { - x } \cdot \left( - x ^ { 2 } + 2 x \right)$. (ii) Determine the global maximum of $w$ by calculation. (3) Given is the function $w _ { \text {new } }$ with the equation
$$w _ { \text {new } } ( x ) = 3 \cdot \mathrm { e } ^ { - ( x - 2 ) } \cdot ( x - 2 ) ^ { 2 } , x \in \mathbb { R } .$$
The graph of $w _ { \text {new } }$ is the locus curve of the inflection points of another family of functions $v _ { k }$ with $k \geq - 0,5$.
Give a possible function equation for $v _ { k }$.

For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$.
Show that for all $n \in \mathbb{N}^*$, $$\sup_{x \in [-1,1]} \left| T_n'(x) \right| = 2^{1-n} n^2$$
Is this supremum attained? If so, specify for which values of $x$.

Let $U$ be the open set of $\mathbb{R}^{2}$ defined by $$U := \left\{(t, x) \in \mathbb{R}^{2} \mid t > 0 \text{ and } x > -t\right\}$$ and let $f$ be the function defined on $U$ by $$f(t, x) = t^{2} \ln\left(1 + \frac{x}{t}\right) - tx.$$
(a) Show that for $(t, x) \in U$, we have $$x \leqslant 0 \Rightarrow f(t, x) \leqslant -\frac{x^{2}}{2}.$$
(b) For $x > 0$, show that we have $$\forall t \geqslant 1, \quad f(t, x) \leqslant f(1, x).$$ For this, one may begin by writing $\frac{\partial f}{\partial t}(t, x)$ in the form $t F(x/t)$ for a certain function $F$ that one will study.

Let $p \in \mathbb{N}^{\star}$. We assume that $f, g_1, \ldots, g_p$ are functions from $\mathbb{R}^n$ to $\mathbb{R}$ differentiable on $\mathbb{R}^n$, and that $$K = \left\{ x \in \mathbb{R}^n, g_1(x) \leqslant 0, \ldots, g_p(x) \leqslant 0 \right\}$$ is non-empty. For all $x \in K$, we denote $I_x = \left\{ i \in \llbracket 1, p \rrbracket, g_i(x) = 0 \right\}$. Show that for all $x \in K$, $$\mathcal{A}_K(x) \subset \left\{ h \in \mathbb{R}^n, \forall i \in I_x, \langle \nabla g_i(x), h \rangle \leqslant 0 \right\}.$$

Let $p \in \mathbb{N}^{\star}$. We assume that $f, g_1, \ldots, g_p$ are functions from $\mathbb{R}^n$ to $\mathbb{R}$ differentiable on $\mathbb{R}^n$, and that $$K = \left\{ x \in \mathbb{R}^n, g_1(x) \leqslant 0, \ldots, g_p(x) \leqslant 0 \right\}$$ is non-empty. For all $x \in K$, we denote $I_x = \left\{ i \in \llbracket 1, p \rrbracket, g_i(x) = 0 \right\}$. We consider $x^{\star} \in K$ and we make the following hypothesis: $$(H) \quad \text{there exists } v \in \mathbb{R}^n \text{ such that for all } i \in I_{x^{\star}}, \langle \nabla g_i(x^{\star}), v \rangle < 0.$$ Show that $\mathcal{A}_K(x^{\star}) = \left\{ h \in \mathbb{R}^n, \forall i \in I_{x^{\star}}, \langle \nabla g_i(x^{\star}), h \rangle \leqslant 0 \right\}$.

Let $p \in \mathbb{N}^{\star}$. We assume that $f, g_1, \ldots, g_p$ are functions from $\mathbb{R}^n$ to $\mathbb{R}$ differentiable on $\mathbb{R}^n$, and that $$K = \left\{ x \in \mathbb{R}^n, g_1(x) \leqslant 0, \ldots, g_p(x) \leqslant 0 \right\}$$ is non-empty. For all $x \in K$, we denote $I_x = \left\{ i \in \llbracket 1, p \rrbracket, g_i(x) = 0 \right\}$. Show that if $x^{\star} \in K$ is such that $(\nabla g_i(x^{\star}))_{i \in I_{x^{\star}}}$ forms a free family, then hypothesis $(H)$ is verified.

Let $p \in \mathbb{N}^{\star}$. We assume that $f, g_1, \ldots, g_p$ are functions from $\mathbb{R}^n$ to $\mathbb{R}$ differentiable on $\mathbb{R}^n$, and that $$K = \left\{ x \in \mathbb{R}^n, g_1(x) \leqslant 0, \ldots, g_p(x) \leqslant 0 \right\}$$ is non-empty. For all $x \in K$, we denote $I_x = \left\{ i \in \llbracket 1, p \rrbracket, g_i(x) = 0 \right\}$. Suppose that $f$ attains at $x^{\star} \in K$ a local minimum on $K$, and that hypothesis $(H)$ is verified. Show that there exist non-negative real numbers $\mu_1^{\star}, \ldots, \mu_p^{\star}$ such that $$\left\{ \begin{array}{l} \nabla f(x^{\star}) + \sum_{i=1}^{p} \mu_i^{\star} \nabla g_i(x^{\star}) = 0 \\ \mu_i^{\star} g_i(x^{\star}) = 0 \text{ for all } i \in \llbracket 1, p \rrbracket. \end{array} \right.$$

Let $p \in \mathbb{N}^{\star}$. We assume that $f, g_1, \ldots, g_p$ are functions from $\mathbb{R}^n$ to $\mathbb{R}$ differentiable on $\mathbb{R}^n$, and that $$K = \left\{ x \in \mathbb{R}^n, g_1(x) \leqslant 0, \ldots, g_p(x) \leqslant 0 \right\}$$ is non-empty. Suppose in this question that the functions $f, g_1, \ldots, g_p$ are convex. Let $x^{\star} \in K$ and $\mu_1^{\star}, \ldots, \mu_p^{\star} \in \mathbb{R}_+$ be such that $$\left\{ \begin{array}{l} \nabla f(x^{\star}) + \sum_{i=1}^{p} \mu_i^{\star} \nabla g_i(x^{\star}) = 0 \\ \mu_i^{\star} g_i(x^{\star}) = 0 \text{ for all } i \in \llbracket 1, p \rrbracket \end{array} \right.$$ is verified. Show that $f$ admits at $x^{\star}$ a global minimum on $K$.

Let $p \in \llbracket 1, n \rrbracket$. We assume that $f, g_1, \ldots, g_p$ are differentiable functions from $\mathbb{R}^n$ to $\mathbb{R}$, that $f$ is $\alpha$-convex for some $\alpha \in \mathbb{R}_+^{\star}$, and that the functions $g_1, \ldots, g_p$ are convex. We further assume that $$K = \left\{ x \in \mathbb{R}^n, g_1(x) \leqslant 0, \ldots, g_p(x) \leqslant 0 \right\}$$ is non-empty. We denote $g(x) = \begin{pmatrix} g_1(x) \\ \vdots \\ g_p(x) \end{pmatrix}$ for all $x \in \mathbb{R}^n$. We introduce the function $\mathcal{L} : \mathbb{R}^n \times \mathbb{R}_+^p \rightarrow \mathbb{R}$ defined by $$\mathcal{L}(x, \mu) = f(x) + \sum_{i=1}^{p} \mu_i g_i(x)$$ for all $x \in \mathbb{R}^n$ and all $\mu = (\mu_1, \ldots, \mu_p) \in \mathbb{R}_+^p$. Show that $\inf_{x \in K} f(x) = \inf_{x \in \mathbb{R}^n} \sup_{\mu \in \mathbb{R}_+^p} \mathcal{L}(x, \mu)$.

Let $p \in \llbracket 1, n \rrbracket$. We assume that $f, g_1, \ldots, g_p$ are differentiable functions from $\mathbb{R}^n$ to $\mathbb{R}$, that $f$ is $\alpha$-convex for some $\alpha \in \mathbb{R}_+^{\star}$, and that the functions $g_1, \ldots, g_p$ are convex. We further assume that $$K = \left\{ x \in \mathbb{R}^n, g_1(x) \leqslant 0, \ldots, g_p(x) \leqslant 0 \right\}$$ is non-empty. We denote $g(x) = \begin{pmatrix} g_1(x) \\ \vdots \\ g_p(x) \end{pmatrix}$ for all $x \in \mathbb{R}^n$. We introduce the function $\mathcal{L} : \mathbb{R}^n \times \mathbb{R}_+^p \rightarrow \mathbb{R}$ defined by $$\mathcal{L}(x, \mu) = f(x) + \sum_{i=1}^{p} \mu_i g_i(x)$$ for all $x \in \mathbb{R}^n$ and all $\mu = (\mu_1, \ldots, \mu_p) \in \mathbb{R}_+^p$. Show that for all $\mu \in \mathbb{R}_+^p$, there exists a unique $x_\mu \in \mathbb{R}^n$ satisfying $\mathcal{L}(x_\mu, \mu) = \inf_{x \in \mathbb{R}^n} \mathcal{L}(x, \mu)$.

Let $p \in \llbracket 1, n \rrbracket$. We assume that $f, g_1, \ldots, g_p$ are differentiable functions from $\mathbb{R}^n$ to $\mathbb{R}$, that $f$ is $\alpha$-convex for some $\alpha \in \mathbb{R}_+^{\star}$, and that the functions $g_1, \ldots, g_p$ are convex. We further assume that $$K = \left\{ x \in \mathbb{R}^n, g_1(x) \leqslant 0, \ldots, g_p(x) \leqslant 0 \right\}$$ is non-empty. We denote $g(x) = \begin{pmatrix} g_1(x) \\ \vdots \\ g_p(x) \end{pmatrix}$ for all $x \in \mathbb{R}^n$. We introduce the function $\mathcal{L} : \mathbb{R}^n \times \mathbb{R}_+^p \rightarrow \mathbb{R}$ defined by $$\mathcal{L}(x, \mu) = f(x) + \sum_{i=1}^{p} \mu_i g_i(x)$$ for all $x \in \mathbb{R}^n$ and all $\mu = (\mu_1, \ldots, \mu_p) \in \mathbb{R}_+^p$. For all $\mu \in \mathbb{R}_+^p$, we denote $G(\mu) := \inf_{x \in \mathbb{R}^n} \mathcal{L}(x, \mu) = \mathcal{L}(x_\mu, \mu)$. We say that $(\bar{x}, \bar{\mu}) \in \mathbb{R}^n \times \mathbb{R}_+^p$ is a saddle point of $\mathcal{L}$ if $$\mathcal{L}(\bar{x}, \bar{\mu}) = \inf_{x \in \mathbb{R}^n} \mathcal{L}(x, \bar{\mu}) \quad \text{and} \quad \mathcal{L}(\bar{x}, \bar{\mu}) = \sup_{\mu \in \mathbb{R}_+^p} \mathcal{L}(\bar{x}, \mu).$$ We assume in this question that $(\bar{x}, \bar{\mu}) \in \mathbb{R}^n \times \mathbb{R}_+^p$ is a saddle point of $\mathcal{L}$. a. Show that $\bar{x}$ is a solution of $(P)$: $\inf_{x \in K} f(x)$. b. Show that $\bar{\mu}$ is a solution of $(Q)$: $\sup_{\mu \in \mathbb{R}_+^p} G(\mu)$. c. Show that $\inf_{x \in \mathbb{R}^n} \sup_{\mu \in \mathbb{R}_+^p} \mathcal{L}(x, \mu) = \sup_{\mu \in \mathbb{R}_+^p} \inf_{x \in \mathbb{R}^n} \mathcal{L}(x, \mu)$.

Let $p \in \llbracket 1, n \rrbracket$. We assume that $f, g_1, \ldots, g_p$ are differentiable functions from $\mathbb{R}^n$ to $\mathbb{R}$, that $f$ is $\alpha$-convex for some $\alpha \in \mathbb{R}_+^{\star}$, and that the functions $g_1, \ldots, g_p$ are convex. We further assume that $$K = \left\{ x \in \mathbb{R}^n, g_1(x) \leqslant 0, \ldots, g_p(x) \leqslant 0 \right\}$$ is non-empty. We denote $g(x) = \begin{pmatrix} g_1(x) \\ \vdots \\ g_p(x) \end{pmatrix}$ for all $x \in \mathbb{R}^n$. We introduce the function $\mathcal{L} : \mathbb{R}^n \times \mathbb{R}_+^p \rightarrow \mathbb{R}$ defined by $$\mathcal{L}(x, \mu) = f(x) + \sum_{i=1}^{p} \mu_i g_i(x)$$ for all $x \in \mathbb{R}^n$ and all $\mu = (\mu_1, \ldots, \mu_p) \in \mathbb{R}_+^p$. For all $\mu \in \mathbb{R}_+^p$, we denote $G(\mu) := \inf_{x \in \mathbb{R}^n} \mathcal{L}(x, \mu)$. We consider $x^{\star} \in K$ a solution of $(P)$ satisfying hypothesis $(H)$. Let $\mu^{\star} = (\mu_1^{\star}, \ldots, \mu_p^{\star})$ as in question III.7. Show that $\mu^{\star}$ is a solution of $(Q)$.

Let $p \in \llbracket 1, n \rrbracket$. We assume that $f, g_1, \ldots, g_p$ are differentiable functions from $\mathbb{R}^n$ to $\mathbb{R}$, that $f$ is $\alpha$-convex for some $\alpha \in \mathbb{R}_+^{\star}$, and that the functions $g_1, \ldots, g_p$ are convex. We further assume that $$K = \left\{ x \in \mathbb{R}^n, g_1(x) \leqslant 0, \ldots, g_p(x) \leqslant 0 \right\}$$ is non-empty. We denote $g(x) = \begin{pmatrix} g_1(x) \\ \vdots \\ g_p(x) \end{pmatrix}$ for all $x \in \mathbb{R}^n$. We introduce the function $\mathcal{L} : \mathbb{R}^n \times \mathbb{R}_+^p \rightarrow \mathbb{R}$ defined by $$\mathcal{L}(x, \mu) = f(x) + \sum_{i=1}^{p} \mu_i g_i(x)$$ for all $x \in \mathbb{R}^n$ and all $\mu = (\mu_1, \ldots, \mu_p) \in \mathbb{R}_+^p$. For all $\mu \in \mathbb{R}_+^p$, we denote $G(\mu) := \inf_{x \in \mathbb{R}^n} \mathcal{L}(x, \mu) = \mathcal{L}(x_\mu, \mu)$. We assume throughout this question that the function $\mu \in \mathbb{R}_+^p \mapsto x_\mu$ is continuous. We consider a solution $\bar{\mu} \in \mathbb{R}_+^p$ of $(Q)$. a. Let $\mu \in \mathbb{R}_+^p$ and $\xi \in \mathbb{R}^p$ be such that $\mu + \xi \in \mathbb{R}_+^p$. Show that for all $t \in [0,1]$, $\mu + t\xi \in \mathbb{R}_+^p$, and $$\lim_{\substack{t \rightarrow 0 \\ t > 0}} \frac{G(\mu + t\xi) - G(\mu)}{t} = \langle g(x_\mu), \xi \rangle.$$ Deduce that for all $\mu \in \mathbb{R}_+^p$, $\langle g(x_{\bar{\mu}}), \mu - \bar{\mu} \rangle \leqslant 0$. b. Show that $x_{\bar{\mu}}$ is a solution of $(P)$.

We denote by $\widehat{S}$ the set of $f \in S_*$ satisfying $\lim_{x \rightarrow -\infty} f(x) = -\infty$ and $\lim_{x \rightarrow +\infty} f(x) = +\infty$. We denote by $\operatorname{Mi}(f)$ the set of minima of $f$ and by $\operatorname{Ma}(f)$ the set of maxima of $f$, so $E(f) = \operatorname{Mi}(f) \cup \operatorname{Ma}(f)$.
1. Let $f \in \widehat{S}$. a. Verify that $\operatorname{Card} \operatorname{Mi}(f) = \operatorname{Card} \operatorname{Ma}(f)$ and that for $y \in \mathbb{R}$, $f^{-1}([-\infty, y])$ is the union of non-empty open intervals that are pairwise disjoint. We denote by $\mathscr{I}(y)$ their set. b. Show that for every element $M$ of $\operatorname{Ma}(f)$, there exists a unique element $m$ of $\operatorname{Mi}(f)$ such that $f(m) < f(M)$ and $m > M$.

Let $a < b$ be two real numbers and $f : [ a , b ] \rightarrow \mathbb { R }$ be an infinitely differentiable function. Let us call (H) the following hypothesis: there exists a unique point $x _ { 0 } \in [ a , b ]$ where $f$ attains its maximum, we have $a < x _ { 0 } < b$, and $f ^ { \prime \prime } \left( x _ { 0 } \right) \neq 0$.
Show that under hypothesis $( H )$, we have $f ^ { \prime \prime } \left( x _ { 0 } \right) < 0$.

Let $a < b$ be two real numbers and $f : [ a , b ] \rightarrow \mathbb { R }$ be an infinitely differentiable function. Let us call (H) the following hypothesis: there exists a unique point $x _ { 0 } \in [ a , b ]$ where $f$ attains its maximum, we have $a < x _ { 0 } < b$, and $f ^ { \prime \prime } \left( x _ { 0 } \right) \neq 0$.
Under hypothesis $( \mathrm { H } )$, show that for all $\delta > 0$ such that $\delta < \min \left( x _ { 0 } - a , b - x _ { 0 } \right)$, we have the asymptotic equivalence, as $t \rightarrow + \infty$, $$\int _ { a } ^ { b } e ^ { t f ( x ) } \mathrm { d } x \sim \int _ { x _ { 0 } - \delta } ^ { x _ { 0 } + \delta } e ^ { t f ( x ) } \mathrm { d } x$$

Let $a < b$ be two real numbers and $f : [ a , b ] \rightarrow \mathbb { R }$ be an infinitely differentiable function. Let us call (H) the following hypothesis: there exists a unique point $x _ { 0 } \in [ a , b ]$ where $f$ attains its maximum, we have $a < x _ { 0 } < b$, and $f ^ { \prime \prime } \left( x _ { 0 } \right) \neq 0$.
Under hypothesis (H), show the asymptotic equivalence, as $t \rightarrow + \infty$, $$\int _ { a } ^ { b } e ^ { t f ( x ) } \mathrm { d } x \sim e ^ { t f \left( x _ { 0 } \right) } \sqrt { \frac { 2 \pi } { t \left| f ^ { \prime \prime } \left( x _ { 0 } \right) \right| } }$$

From now on, $f$ denotes an infinitely differentiable function from $[0,1]$ to $\mathbb{R}$. We assume that there exists a unique point $x_0 \in [0,1]$ where $f'$ vanishes. We also assume that $f''(x_0) > 0$.
For all $x \in [x_0, 1]$, we define $$h(x) = \sqrt{|f(x) - f(x_0)|}$$
(a) Show that the function $h$ defines a bijection from $[x_0, 1]$ to $[0, h(1)]$.
(b) Show that the application $h$ is differentiable at $x_0$ from the right, and that $h'_+(x_0) = \sqrt{\frac{f''(x_0)}{2}}$.

From now on, $f$ denotes an infinitely differentiable function from $[ 0,1 ]$ to $\mathbb { R }$. We assume that there exists a unique point $x _ { 0 } \in \left[ 0,1 \left[ \right. \right.$ where $f ^ { \prime }$ vanishes. We also assume that $f ^ { \prime \prime } \left( x _ { 0 } \right) > 0$. We are also given an infinitely differentiable function $g : [ 0,1 ] \rightarrow \mathbb { R }$.
For all $x \in \left[ x _ { 0 } , 1 \right]$, we define $$h ( x ) = \sqrt { \left| f ( x ) - f \left( x _ { 0 } \right) \right| }$$
(a) Show that the function $h$ defines a bijection from $\left[ x _ { 0 } , 1 \right]$ to $[ 0 , h ( 1 ) ]$.
(b) Show that the map $h$ is differentiable at $x _ { 0 }$ on the right, and that $h ^ { \prime } \left( x _ { 0 } \right) = \sqrt { \frac { f ^ { \prime \prime } \left( x _ { 0 } \right) } { 2 } }$.

From now on, $f$ denotes an infinitely differentiable function from $[ 0,1 ]$ to $\mathbb { R }$. We assume that there exists a unique point $x _ { 0 } \in \left[ 0,1 \left[ \right. \right.$ where $f ^ { \prime }$ vanishes. We also assume that $f ^ { \prime \prime } \left( x _ { 0 } \right) > 0$. We are also given an infinitely differentiable function $g : [ 0,1 ] \rightarrow \mathbb { R }$.
For all $x \in \left[ x _ { 0 } , 1 \right]$, we define $$h ( x ) = \sqrt { \left| f ( x ) - f \left( x _ { 0 } \right) \right| }$$ We admit that the bijection $$h : \left\{ \begin{array} { c c c } { \left[ x _ { 0 } , 1 \right] } & \rightarrow & { [ 0 , h ( 1 ) ] } \\ x & \mapsto & h ( x ) \end{array} \right.$$ admits an inverse map $h ^ { - 1 } : [ 0 , h ( 1 ) ] \rightarrow \left[ x _ { 0 } , 1 \right]$ that is infinitely differentiable.
Show that, as $t \rightarrow + \infty$, $$\int _ { x _ { 0 } } ^ { 1 } \sin ( t f ( x ) ) \mathrm { d } x = \sin \left( t f \left( x _ { 0 } \right) + \frac { \pi } { 4 } \right) \sqrt { \frac { \pi } { 2 t f ^ { \prime \prime } \left( x _ { 0 } \right) } } + O \left( \frac { 1 } { t } \right)$$