Answer:
Step-by-step explanation:The statement "There exists a value c in the open interval -1 < x < 2 such that f'(c) = -4" is true.
To determine the validity of the statement, we need to check if there exists a value c in the open interval -1 < x < 2 such that f'(c) = -4.
Since the function f is continuous on the closed interval -1 ≤ x ≤ 2 and differentiable on the open interval -1 < x < 2, the function satisfies the conditions for the Mean Value Theorem.
According to the Mean Value Theorem, if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one value c in the open interval such that f'(c) = (f(b) - f(a))/(b - a), where a and b are the endpoints of the closed interval.
In this case, the closed interval is -1 ≤ x ≤ 2, so a = -1 and b = 2.
Using the values from the table, we can calculate (f(b) - f(a))/(b - a):
(f(2) - f(-1))/(2 - (-1)) = (1 - 9)/(2 + 1) = -8/3
Since -8/3 is not equal to -4, we can conclude that the statement "There exists a value c in the open interval -1 < x < 2 such that f'(c) = -4" is false.
