Step 1 :The Mean Value Theorem states that if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one number c in the interval (a, b) such that the derivative of the function at that point is equal to the average rate of change of the function over the interval [a, b].
Step 2 :The average rate of change of the function over the interval [a, b] is given by the formula: \[f'(c) = \frac{f(b) - f(a)}{b - a}\]
Step 3 :So, we need to find a point c in the interval (-8, -2) such that the derivative of the function at that point is equal to the average rate of change of the function over the interval [-8, -2].
Step 4 :First, we need to find the derivative of the function f(x) = 2 - 10/(7x).
Step 5 :Then, we need to find the average rate of change of the function over the interval [-8, -2].
Step 6 :Finally, we need to solve the equation f'(c) = (f(-2) - f(-8))/(-2 - (-8)) for c.
Step 7 :The solution to the equation f'(c) = avg_rate_of_change for c is [-4, 4]. However, only -4 is in the interval (-8, -2). Therefore, the point c that satisfies the Mean Value Theorem is -4.
Step 8 :Final Answer: \[c=\boxed{-4}\]