The department of transportation has proposed a new 7-mile stretch of highway. The elevation of the highway x miles along the path is given by E(x)=.004 (x^4-8x^3)+2. Where is the steepest point

Answer 1

that's where E' is a max

E' = .004(4x^3-24x^2) = .008(x^3-6x^2)
E' is max where E" = 0
E" = .008(3x^2-12x) = .024x(x-4)
so, |E'| is max when x=4, as you can see at

http://www.wolframalpha.com/input/?i=.004+%28x%5E4-8x%5E3%29%2B2