Throughout this part, $\lambda$ is a real number belonging to the interval $]0,1[$ and $f, g, h$ are functions in $C^{0}(\mathbb{R}, \mathbb{R}_{+})$ that are integrable and satisfy the following inequality $$\forall x \in \mathbb{R}, \forall y \in \mathbb{R}, \quad h(\lambda x + (1-\lambda) y) \geq f(x)^{\lambda} g(y)^{1-\lambda}.$$ In questions 3), 4) and 5) we additionally assume that $f$ and $g$ are strictly positive, that is, for all real $x$, $f(x) > 0$ and $g(x) > 0$.
Show that the image set of the application $w$ defined on $]0,1[$ by $$\forall t \in ]0,1[, \quad w(t) = \lambda u(t) + (1-\lambda) v(t),$$ is equal to $\mathbb{R}$. Then prove that $w$ defines a change of variable from $]0,1[$ to $\mathbb{R}$. Using this and $\int_{-\infty}^{+\infty} h(w)\,dw$, show that $f$, $g$ and $h$ satisfy the "P-L" inequality $$\int_{-\infty}^{+\infty} h(x)\,dx \geq \left(\int_{-\infty}^{+\infty} f(x)\,dx\right)^{\lambda} \left(\int_{-\infty}^{+\infty} g(x)\,dx\right)^{1-\lambda}.$$

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 $f''(x_0) \neq 0$.
We admit the identity $\int_{-\infty}^{+\infty} \exp(-x^2) \mathrm{d}x = \sqrt{\pi}$.
Under hypothesis (H), show the asymptotic equivalence, as $t \rightarrow +\infty$, $$\int_a^b e^{tf(x)} \mathrm{d}x \sim e^{tf(x_0)} \sqrt{\frac{2\pi}{t|f''(x_0)|}}$$

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 (H), show the asymptotic equivalence, as $t \rightarrow + \infty$, $$\int _ { a } ^ { b } e ^ { t f ( x ) } \mathrm { d } x \sim e ^ { t f \left( x _ { 0 } \right) } \sqrt { \frac { 2 \pi } { t \left| f ^ { \prime \prime } \left( x _ { 0 } \right) \right| } }$$

Let $r$ be the function defined by
$$\begin{aligned} r : ] - \pi ; \pi [ & \longrightarrow \mathbf { C } \\ \theta & \longmapsto \int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t e ^ { \mathrm { i } \theta } } \mathrm {~d} t \end{aligned}$$
where $x$ is a fixed element of $]0;1[$. Show that the function $r$ is of class $\mathcal { C } ^ { 1 }$ on $] - \pi ; \pi [$ and that:
$$\forall \theta \in ] - \pi ; \pi \left[ , \quad r ^ { \prime } ( \theta ) = - \mathrm { i } e ^ { \mathrm { i } \theta } \cdot \int _ { 0 } ^ { + \infty } \frac { t ^ { x } } { \left( 1 + t \mathrm { e } ^ { \mathrm { i } \theta } \right) ^ { 2 } } \mathrm {~d} t . \right.$$
Hint: let $\beta \in ] 0 ; \pi [$, show that for all $\theta \in [ - \beta ; \beta ]$ and $t \in [ 0 , + \infty [$, $\left| 1 + t e ^ { i \theta } \right| ^ { 2 } \geq \left| 1 + t e ^ { i \beta } \right| ^ { 2 } = ( t + \cos ( \beta ) ) ^ { 2 } + ( \sin ( \beta ) ) ^ { 2 }$.

In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
Deduce that $$Z_n(h) = \sqrt{\frac{n}{2\mathrm{e}^{\beta}\pi\beta}} \int_{-\infty}^{+\infty} \left(\sum_{x \in \Lambda_n} \mathrm{e}^{(t+h)s_n(x)}\right) \mathrm{e}^{-\frac{nt^2}{2\beta}} \mathrm{~d}t$$

In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$.
Show that $$Z_n(h) = \sqrt{\frac{n}{2\mathrm{e}^{\beta}\pi\beta}} \int_{-\infty}^{+\infty} \mathrm{e}^{-nG_h(x)} \mathrm{d}x$$

In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$. We now assume that $h > 0$.
We denote $\widehat{G}_h : x \longmapsto G_h(x + u_h) - \min G_h$. Show that $$\psi_n(h) = -G_h(u_h) - \frac{1}{2n}\ln\left(2\mathrm{e}^{\beta}\pi\beta\right) + \frac{1}{n}\ln\left(\int_{-\infty}^{+\infty} \mathrm{e}^{-n\widehat{G}_h\left(\frac{t}{\sqrt{n}}\right)} \mathrm{d}t\right)$$

3. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \\ \text { MATHEMATICS \& COMPUTER SCIENCE } \end{array} \right\}$ ONLY.

Computer Science and Computer Science \& Philosophy applicants should turn to page 16.
In this question we shall investigate when functions are close approximations to each other. We define $| x |$ to be equal to $x$ if $x \geqslant 0$ and to $- x$ if $x < 0$. With this notation we say that a function $f$ is an excellent approximation to a function $g$ if
$$| f ( x ) - g ( x ) | \leqslant \frac { 1 } { 320 } \quad \text { whenever } \quad 0 \leqslant x \leqslant \frac { 1 } { 2 }$$
we say that $f$ is a good approximation to a function $g$ if
$$| f ( x ) - g ( x ) | \leqslant \frac { 1 } { 100 } \quad \text { whenever } \quad 0 \leqslant x \leqslant \frac { 1 } { 2 }$$
For example, any function $f$ is an excellent approximation to itself. If $f$ is an excellent approximation to $g$ then $f$ is certainly a good approximation to $g$, but the converse need not hold.
(i) Give an example of two functions $f$ and $g$ such that $f$ is a good approximation to $g$ but $f$ is not an excellent approximation to $g$.
(ii) Show that if
$$f ( x ) = x \quad \text { and } \quad g ( x ) = x + \frac { \sin \left( 4 x ^ { 2 } \right) } { 400 }$$
then $f$ is an excellent approximation to $g$. For the remainder of the question we are going to a try to find a good approximation to the exponential function. This function, which we shall call $h$, satisfies the following equation
$$h ( x ) = 1 + \int _ { 0 } ^ { x } h ( t ) \mathrm { d } t \quad \text { whenever } \quad x \geqslant 0$$
You may not use any other properties of the exponential function during this question, and any attempt to do so will receive no marks.
Let
$$f ( x ) = 1 + x + \frac { x ^ { 2 } } { 2 } + \frac { x ^ { 3 } } { 6 }$$
(iii) Show that if
$$g ( x ) = 1 + \int _ { 0 } ^ { x } f ( t ) \mathrm { d } t$$
then $f$ is an excellent approximation to $g$.
(iv) Show that for $x \geqslant 0$
$$h ( x ) - f ( x ) = g ( x ) - f ( x ) + \int _ { 0 } ^ { x } ( h ( t ) - f ( t ) ) \mathrm { d } t$$
(v) You are given that $h ( x ) - f ( x )$ has a maximum value on the interval $0 \leqslant x \leqslant 1 / 2$ at $x = x _ { 0 }$. Explain why
$$\int _ { 0 } ^ { x } ( h ( t ) - f ( t ) ) \mathrm { d } t \leqslant \frac { 1 } { 2 } \left( h \left( x _ { 0 } \right) - f \left( x _ { 0 } \right) \right) \quad \text { whenever } \quad 0 \leqslant x \leqslant \frac { 1 } { 2 }$$
(vi) You are also given that $f ( x ) \leqslant h ( x )$ for all $0 \leqslant x \leqslant \frac { 1 } { 2 }$. Show that $f$ is a good approximation to $h$ when $0 \leqslant x \leqslant \frac { 1 } { 2 }$.
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