Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a function of class $C^{\infty}$ on $\mathbb{R}$ and 1-periodic. Let $t \in [-1/2, 1/2]$. We consider the function $G_{t}$ defined on $[-1/2, 1/2]$ by
$$\forall x \in \left[-\frac{1}{2}, \frac{1}{2}\right], \quad G_{t}(x) = f'(x+t)\sin(\pi x) - (f(x+t)-f(t))\pi\cos(\pi x)$$
Establish the existence of a real number $D$, independent of $x$ and $t$, such that
$$\forall t \in \left[-\frac{1}{2}, \frac{1}{2}\right], \quad \forall x \in \left[-\frac{1}{2}, \frac{1}{2}\right], \quad |G_{t}(x)| \leqslant D x^{2}$$

Let $g : \mathbb{R}_+ \rightarrow \mathbb{R}$ be the function defined by $g(x) = \ln\left(1 - p + pe^x\right)$ for all $x \geq 0$.
a. Show that $g$ is well defined and of class $C^2$ on $\mathbb{R}_+$. For $x \geq 0$, express $g''(x)$ in the form $\frac{\alpha\beta}{(\alpha+\beta)^2}$ where $\alpha$ and $\beta$ are positive reals that may depend on $x$.
b. Show that $g''(x) \leq \frac{1}{4}$ for all $x \geq 0$.
c. Show that $$\ln\left(1 - p + pe^x\right) \leq px + \frac{x^2}{8} \text{ for all } x \geq 0$$

Let $A \in S_n^{++}(\mathbf{R})$ and let $f : t \mapsto \ln(\operatorname{det}(I_n + tA))$. Show that $$\forall t \in \mathbf{R}_+, \quad \ln(\operatorname{det}(I_n + tA)) \leq \operatorname{Tr}(A) t.$$

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and let $f : t \mapsto \ln \left( \operatorname { det } \left( I _ { n } + t A \right) \right)$. Show that
$$\forall t \in \mathbf { R } _ { + } , \quad \ln \left( \operatorname { det } \left( I _ { n } + t A \right) \right) \leq \operatorname { Tr } ( A ) t$$