Para determinar pontos de Móximo locais, Mínimo locais e ponto d sela de uma funçáo de duas variáveis reais \( z=f(x, y) \) devemos primeiramente encontrar os pontos críticos da funçao.Os pontos críticos também chamados de pontos estacionários que sôo ponts nos quais a funçáo \( f(x, y) \) tem Escolha uma opção: a. \( \frac{\partial f}{\partial x}=0 \). b. \( \frac{\partial f}{\partial y}=0 \) c. \( \frac{\partial f}{\partial x}=0 \) e \( \frac{\partial f}{\partial y}=0 \) d. \( \frac{\partial f}{\partial x} \neq 0 \) e \( \frac{\partial f}{\partial y} \neq 0 \) e. \( \frac{\partial f}{\partial x}<0 \) e \( \frac{\partial f}{\partial y}>0 \)

On prend pour première approximation de \( \alpha \) l'abscisse \( x_{1} \) du point d'in- tersection de la tangente au point \( (b, f(b)) \) avec l'axe \( (O x) \), puis on itère la procédure sur \( \left[a, x_{1}\right] \) pour obtenir \( \mathrm{x}_{2} \ldots \) 2. Montrer que \( x_{1}=b-\frac{f(b)}{f^{\prime}(b)} \). 3. Montrer que \( \frac{\mathrm{f}(\mathrm{b})-\mathrm{f}\left(\mathrm{x}_{1}\right)}{\mathrm{b}-\mathrm{x}_{1}} \leq \mathrm{f}^{\prime}(\mathrm{b}) \). 4. En déduire que \( f\left(x_{1}\right) \geq 0 \), puis \( x_{1} \geq \alpha \). \( \quad \) Ceci nous ramène à considérer la suite définie par \( \mathrm{x}_{0}=\mathrm{b} \) et la relation: \( \forall \mathrm{n} \in \mathbb{N} \mathrm{x}_{\mathrm{n}+1}=\mathrm{x}_{\mathrm{n}}-\frac{\mathrm{f}\left(\mathrm{s}_{\mathrm{n}}\right)}{\mathrm{f}^{\prime}\left(\mathrm{x}_{\mathrm{n}}\right)} \) En introduisant la fonction \( g: x \rightarrow x-\frac{f(x)}{f \prime(x)} \) on peut écrire \( x_{n+1}= \) \( g\left(x_{n}\right) \). 5. Etudier la monotonie de g sur \( [\mathrm{a}, \mathrm{b}] \) et montrer que \( \mathrm{g}([\alpha, \mathrm{b}]) \subset[\alpha, \mathrm{b}] \).

Find the derivative of the function. \[ y=\tanh \left(6 x^{2}-7\right) \] \( y^{\prime}=\square \)

Evaluate the function. If the value is not a rational number, rouna (a) \( \operatorname{csch}^{-1}(3) \) (b) \( \operatorname{coth}^{-1}(6) \)

Find an equation of the tangent line to the graph of the function at the point \( (1,1) \). \[ y=x^{\cosh (x)} \] \( y=\square \)

Consider the following. \[ y=\sinh \left(4-x^{2}\right),(2,0) \] Find \( y^{\prime} \) Find the slope of the line tangent to the graph of \( y \) at the given point. Find an equation of the tangent line to the graph of the function at the given point. .