25. Determine the intervals on which the following function is concave up or concave down. Identify any inflection points. \( f(x)=5 x^{4}-50 x^{3}+10 \) Determine the intervals on which the following functions are concave up or concave down. Select the correct choice below and fill in the answer box(es) to complete your choice. (Simplify your answer. Type your answer in interval notation. Use a comma to separate answers as needed.) A. The function is concave up on B. The function is concave up on C. The function is concave down on Locate any inflection points of f . Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. An inflection point occurs at \( x=\square \) B. There are no inflection points for f .

\( f ( x ) = \int | x | d x \)

Partie II- Comportement à l'infini des solutions de \( :(E): y^{\prime \prime}-\frac{1}{1+t^{2}} y^{\prime}+y=0 \) Soi \( f \) une solution non nulle de \( (E): y^{\prime \prime}-\frac{1}{1+t^{2}} y^{\prime}+y=0 \). On note \( q: t \in \mathbb{R}^{+} \mapsto \frac{1}{1+t^{2}} \). 3. Justifier à l'aide du théorème de Cauchy-Lipschitz, que pour tout \( t \in \mathbb{R},\left(f(t), f^{\prime}(t)\right) \neq(0,0) \). 4. En déduire qu'il existe deux applications \( r \in \mathcal{C}^{1}\left(\mathbb{R}^{+}, \mathbb{R}_{+}^{*}\right) \) et \( \theta \in \mathcal{C}^{1}\left(\mathbb{R}^{+}, \mathbb{R}\right) \) telles que \[ \forall t \in \mathbb{R}^{+}, f(t)=r(t) \cos (\theta(t)) \text { et } f^{\prime}(t)=r(t) \sin (\theta(t)) \text {. } \] 5. Exprimer \( r \) en fonction de \( f \) et \( f^{\prime} \). Les fonctions \( r \) et \( \theta \) sont fixées ainsi pour la suite de la partie. (a) Montrer que \( \theta^{\prime}=-1+q \sin (\theta) \cos (\theta) \) (b) Montrer que \( r^{\prime}=q r \sin 2 \). En déduire la monotonie de \( r \). (c) Etudier les variations de \( t \mapsto r(t) \exp (-\arctan (t)) \) et en déduire que \( 0 \leq r(t) \leq r(0) \exp (\operatorname{arctar} \) que \( r \) a une limite strictement positive en \( +\infty \). Démontrer que \( f \) et \( f^{\prime} \) sont bornées .

2. find Maclurin expansion of (2 Points) * \[ \begin{array}{l}f(x)=\sinh x \\ f(x)=1+\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\ldots \ldots \\ f(x)=x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\ldots \ldots \\ f(x)=x+\frac{x^{3}}{3}+\frac{x^{5}}{5}+\ldots \ldots \\ f(x)=x+\frac{x^{3}}{3!}+\frac{x^{5}}{5!}+\ldots\end{array} \]

49) \( y=\sqrt{x} \quad f^{(3)} \)

3. find taylor series of (2 Points) * \( \begin{array}{l}f(x)=e^{x} \cos x \quad \text { at } x=0 \\ f(x)=1+x-\frac{2 x^{3}}{3!}+\ldots \\ f(x)=1+x-\frac{x^{3}}{3!}+\ldots \\ f(x)=1+x-\frac{x^{2}}{2!}+\ldots \\ f(x)=1+x-\frac{2 x^{2}}{2!}+\ldots\end{array} \)