Consider the function f(x)= (-3x^3)-(3x^2)+(2x)-2 Find the average slope of this function on the interval (1,2).   By the Mean Value Theorem, we know there exists a in the open interval (1,2) such that f'(c) is equal to this mean slope. Find the value of c in the interval which works

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You need to use mean value theorem over (1,2) such that:

`f'(c) = (f(2)-f(1))/(2-1)`

You need to evaluate f(2) and f(1), such that:

`f(2) = -24 - 12 + 4 -2`

`f(2) = -34`

`f(1) = -3 - 3 + 2 - 2 = -6`

You need to substitute -34 for f(2) and -6 for f(1) in `f'(c) = (f(2)-f(1))/(2-1)`  such that:

`f'(c) = (-34-6)/1 = -40`

You need to evaluate f'(c), hence you need to find f'(x) such that:

`f'(x) = -9x^2 - 6x + 2`

Substituting c for x in f'(x) yields:

`f'(c) = -9c^2 - 6c + 2`

Hence, evaluating the mean slope value yields`f'(c) = -40`  for any `c in (1,2).`

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