consider the function f(x)=x^3 - x^2 - 3x -2. find the average slope of this function on the interval (-2,3). by the mean value theorem, we know there exists a c in the open interval (-2,3) such that f'(c) is equal to this mean slope. find the two values of c in the interval which work.

Answer 1

f(-2) = -8
f(3) = 7

average slope = (7+8)/(3+2) = 3

f'(x) = 3x^2 - 2x - 3
f'(c) = 3c^2 - 2c - 3

3c^2 - 2c - 3 = 3
3c^2 - 2c - 6 = 0
c = (2 ± √73)/6
= (2 ± 6√2)/6
= 1/3± √2