8. \( f(x)=x^{3}+5 \)

Did you know that polynomial functions like \( f(x) = x^3 + 5 \) have been studied since ancient times? The cubic function, which this equation represents, was pivotal in the development of algebra and calculus! These functions can be used to model a variety of real-world phenomena, from predicting the volume of a cube to modeling cubic growth in population studies. When analyzing the function, remember that it has one local extremum, which is a global minimum in this case since it increases indefinitely as \( x \) approaches both positive and negative infinity. A common mistake is misidentifying the nature of its roots; \( f(x) = 0 \) has one real root and two complex roots due to its cubic nature, which is a key feature in understanding its graph!