Consider the following function...

Question

Consider the function

f(x) = 3x^3 - x^2 +4x +1

Find the average rate of change of this function on the interval (-2,-1)_______ . By the Mean Value Theorem, we know there exists a c in the open interval (-2,-1) such that f'(c) is equal to this average rate of change. Find the value of c in the interval which works_______

Answer 1

Mean Value Theorem: f'(c) = [f(b) - f(a)} ⁄[b-a]

f(-2) = -35, f(1) = 7

Thus, 9x² - 2x + 4 = (7 + 35) ⁄ (1 + 2)

9x²- 2x + 4 = 14

9x² - 2x -10 = 0

x = [2 ± √364] ⁄ 18

(2 + 2√91) ⁄ 18 is not in the given interval

Thus, x = (2 - 2√91) ⁄ 18 ≈ - 0.9488 is the answer