Ejercicio 13-c, TP4, UBAXXI

$\text{13)-c) Determiná}\, x \in [0;5\pi] \text{ tal que }f(x)=\sqrt{2}\text{; si }f(x)=2sen\left(\frac{x}{4}+ \pi\right)$
$$f(x)=\sqrt{2}\quad\Leftrightarrow\quad 2sen\left(\frac{x}{4}+ \pi\right)=\sqrt{2}\\
f(x)=\sqrt{2}\quad\Leftrightarrow\quad 2sen\left(\frac{x}{4}+ \pi\right)= \sqrt{2}\\
f(x)=\sqrt{2}\quad\Leftrightarrow\quad sen\underset{z}{\underbrace{\left(\frac{x}{4}+ \pi\right)}}= \frac{\sqrt{2}}{2}$$

• z $$z=\left(\frac{x}{4}+ \pi \right)\quad \wedge \quad z \in{\mathbb{R}}\\
  \Rightarrow \quad sen(z)=  \frac{\sqrt{2}}{2}\\
  \Rightarrow \quad z_1= \frac{\pi}{4}\quad \wedge \quad z_2=\frac{3\pi}{4}$$

• z₁ $$z_1=\frac{\pi}{4} \quad \Rightarrow \quad \left(\frac{x}{4}+ \pi\right)= \frac{\pi}{4}\\
  \frac{x}{4}=\frac{\pi}{4}- \pi\\
  x= -\frac{3\pi}{4}\cdot 4 \; \therefore\, x=-3\pi\\
  \Rightarrow \; x_1=-3\pi+2k\pi \quad (k \in{\mathbb{Z}})$$

•  z₂$$z_2=\frac{3\pi}{4} \quad \Rightarrow \quad \left(\frac{x}{4}+ \pi\right)= \frac{3\pi}{4}\\
  \frac{x}{4}= \frac{3\pi}{4}-\pi\\
  x= -\frac{\pi}{4}\cdot 4 \; \therefore\, x=-\pi\\
  \Rightarrow \; x_2=-\pi+2k\pi \quad (k \in{\mathbb{Z}})$$

• Período T:$$T=\frac{2k\pi}{\left| a \right|}\; \wedge\; f(x)=2sen\left(\frac{x}{4}+\pi \right)$$
  $$|a|=\frac{1}{4} \quad \Rightarrow \quad T=\frac{2k\pi}{\frac{1}{4}}=4\cdot\frac{2k\pi}{1}=8k\pi$$
  $$\Rightarrow \quad x_1=-3\pi+8k\pi \quad \wedge \quad x_2=-\pi+8k\pi \quad (k \in{\mathbb{Z}})$$
  $$x_2 < 0 \; \wedge \; x_2>5\pi \text{ para cualquier valor de k.} $$

• Valores de k: $$k \in [0;5\pi]\\
  \begin{matrix}
  & k & = & 1 & & &\\
  & & x_1 & = & -3\pi + 8\cdot(1)\cdot \pi & = & 5\pi
  \end{matrix}
  $$
  $$x \in{[0;5\pi]}/f(x)=\sqrt{2}:\; \underline{S=\{5\pi\}}$$