1) \( f(x)=r x^{r}+r x \quad x=1 \)

Did you know that the function you’ve provided can showcase how quickly growth can explode when parameters are adjusted? When you evaluate \( f(1) \), you're essentially finding a mix of constants and a linear term, which gives a fascinating glimpse into polynomial behavior. You get \( f(1) = r \cdot 1^r + r \cdot 1 = 2r \). This means for every unit increase in \( r \), you're doubling it in your result. This kind of function has real-world applications in fields such as physics and economics, where you might model phenomena like population growth or profit strategies. By analyzing how \( r \) interacts with \( x \), you can predict how changes in input will affect outputs, making it a valuable tool for decision-making and forecasting in various industries!