1- Montrer que : \( \exists!\alpha \in] \frac{-1}{2} ; 0\left[/ \sqrt{\alpha+1}=\frac{-2 \alpha}{\alpha+1}\right. \) 2- On considère la fonction \( h \) de la variable réel \( x \) définie sur \( \mathbb{R} \) par: \( h(x)=x^{3}-x^{2}+3 x+1 \). a- Vérifier que : \( h(\alpha)=0 \). b- Montrer que \( \alpha \) est la seule et unique solution de l'équation: \( h(x)=0 \). c- Montrer que \( h \) admet une fonction réciproque \( h^{-1} \) définie sur un intervalle \( J \) que l'on précisera. d- Montrer que : \( \left(\forall x \in \mathbb{R}^{-}\right)\left(h^{-1}(x)<0\right) \)

Numa circunferência, o diâmetro mede 2,88 m. Quanto mede, em decímetro, o raio dessa circunferência?

Vocabulary How do you know if an equation represents a proportional relationship? Use the drop-down menu to show your answer. \begin{tabular}{ll} If an equation can be written in the form \\ \( \qquad \begin{array}{l}\text { Choose.... } \\ y=k+x \\ y=k \\ y=k x\end{array} \) \\ \hline\end{tabular}

1. A discrete \( \boldsymbol{r} \cdot \boldsymbol{\mathrm { X }} \) takes the values 1,2 or 3 and its cumulative distribution function is as follows: \[ \boldsymbol{F}(\boldsymbol{x})=\frac{(\boldsymbol{x}+\boldsymbol{k})^{2}}{16}, \boldsymbol{x}=1,2,3 \] Find the: (i) value of \( \boldsymbol{k} \) where \( \boldsymbol{k}>\mathbf{0} \). (ii) probability distribution of \( \boldsymbol{X} \). (iii) mean and variance of \( \boldsymbol{X} \).

Find \( \frac{d y}{d x} \) using the rules for finding derivatives. \( y=\left(2 x^{2}-3\right)\left(7 x^{2}-9 x+8\right) \) \( \frac{d y}{d x}=\square \)

Find the equation of the circle passing through the three points \( A(-2,5), B(4,9) \) and \( C(2-1) \).