by HeMan
Last Updated August 13, 2018 03:20 AM

In Brin's book "Introduction to dynamical systems", page 10, he defines a cuadratic function $q_\mu$ for $\mu>4$. He observes that $[\frac{1}{\mu}, \frac{1}{2}] \subset q_{\mu}^2([\frac{1}{\mu}, \frac{1}{2}])$ and then he says that by the intermediate value theorem, exists a fixed point of $q_{\mu}^2$, but I don't have any idea about why this is true. Any hint would be appreciated.

$q_\mu^2$ attains maximum at some point of $x_M\in[1/\mu,1/2]$ and minimun at some point $x_m\in[1/\mu,1/2]$.

Since $[1/\mu,1/2]\subset q_\mu^2([1/\mu,1/2])$, then $q_\mu^2(x_M)-x_M\geq1/2-x_M\geq0$ and $q_\mu^2(x_m)-x_m\leq1/mu-x_m\leq0$.

Therefore $q_mu^2(x)-x$ vanishes at some point of $[1/\mu,1/2]$.

- ServerfaultXchanger
- SuperuserXchanger
- UbuntuXchanger
- WebappsXchanger
- WebmastersXchanger
- ProgrammersXchanger
- DbaXchanger
- DrupalXchanger
- WordpressXchanger
- MagentoXchanger
- JoomlaXchanger
- AndroidXchanger
- AppleXchanger
- GameXchanger
- GamingXchanger
- BlenderXchanger
- UxXchanger
- CookingXchanger
- PhotoXchanger
- StatsXchanger
- MathXchanger
- DiyXchanger
- GisXchanger
- TexXchanger
- MetaXchanger
- ElectronicsXchanger
- StackoverflowXchanger
- BitcoinXchanger
- EthereumXcanger