A company manufactures and selv \( x \) televisiuns sets per month. The monthly Cost and price demand Cquations are: \( C(x)=75000+50 x \) and \( P(x)=200-\frac{x}{30} \), \( 0 \leq x \leq 6000 \) Find the max revenue.

Find the average rate of change of \( f(x)=-2 x^{2}-x \) from \( x=2 \) to \( x=6 \), Simplify your answer as much as possible.

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

Consider the following. \[ g(x)=4 x \operatorname{sech}(x) \] Find \( g^{\prime}(x) \). \( g^{\prime}(x)=\square \) Find any critical numbers of \( g \). (Enter your answers as a comma-separated list. Round your answers to four decimal places.) Find the relative extrema of the function. (Round your answers to four decimal places.) relative maxima \( \quad(x, y)=(\square) \) relative minima \[ (x, y)=(\square) \]

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 indefinite integral. (Remember the constant of integration.)