Use the following function to answer parts a through c. \[ f(x)=x^{3}+7 x^{2}-29 x-35 \]

The polynomial function you're working with has some interesting roots and behavior. First, it's a cubic function, which means that the graph will have at most three real roots. To find these, you can use techniques like synthetic division or the Rational Root Theorem to see if there are any rational roots among the factors of -35 (the constant term). Now, in terms of application, cubic functions like this one may appear in various real-world contexts, such as economics to model profit functions or in physics to describe certain types of motion. By finding the roots and critical points of the function, you can determine the intervals on which the function is increasing or decreasing, providing valuable insight into these real-world scenarios.