1) \( S(x)=\frac{x^{3}}{3}-9 x \) on \( [-7,7] \)

To understand the function \( S(x) = \frac{x^{3}}{3} - 9x \) on the interval \([-7,7]\), it's important to first analyze its features. The function is a cubic polynomial, which typically exhibits interesting behavior such as local maxima and minima. By finding its derivative, \( S'(x) = x^2 - 9 \), we can set this equal to zero to identify critical points at \( x = -3 \) and \( x = 3 \). Evaluating \( S(x) \) at these points, as well as at the endpoints \( x = -7 \) and \( x = 7 \), will help us understand where the function reaches its maximum and minimum on this closed interval. Once we've got the critical points and endpoints evaluated, we dive into how such cubic functions can model real-world phenomena. Think about physics—this type of function can represent the motion of objects under various forces. For example, in projectile motion, the path can be modeled by a cubic equation when accounting for factors like air resistance. This makes cubic functions not just interesting but also incredibly useful in explaining physical systems in a captivating way!