Consider the function $\displaystyle {f{{\left({x}\right)}}}={2}{x}^{{3}}-{12}{x}^{{2}}-{72}{x}+{3}$ on the interval $\displaystyle {\left[-{5},{10}\right]}$. Find the average or mean slope of the function on this interval.
Preview Question 6 Part 1 of 3  

By the Mean Value Theorem, we know there exists a $\displaystyle {c}$ in the open interval $\displaystyle {\left(-{5},{10}\right)}$ such that $\displaystyle {f}'{\left({c}\right)}$ is equal to this mean slope. For this problem, there are two values of $\displaystyle {c}$ that work.

The smaller one is Preview Question 6 Part 2 of 3  

and the larger one is Preview Question 6 Part 3 of 3