Final answer:
The Mean Value Theorem can be applied to find the value c in the interval (36, 64) for the function f(x) = 6 * sqrt(x) - 5. However, after calculating, the result does not match any of the provided options, suggesting there may be an error in the question.
Step-by-step explanation:
The Mean Value Theorem (MVT) states that for a continuous and differentiable function f(x) on the closed interval [a, b], there exists at least one number c in the open interval (a, b) such that f'(c) is equal to the average rate of change of the function over [a, b]. This can be written as:
f'(c) = [f(b) - f(a)] / (b - a).
We are given the function f(x) = 6 √ x - 5 on the interval (36, 64). Firstly, we calculate the average rate of change:
Average rate of change = [f(64) - f(36)] / (64 - 36) = [(6√64 - 5) - (6√36 - 5)] / 28 = (6×8 - 6×6) / 28 = 24 / 28 = 6 / 7.
Next, we find the derivative of f(x):
f'(x) = d/dx [6 √ x - 5] = (6/2)×1/x^(1/2) = 3/x^(1/2).
We then set the derivative equal to the average rate of change:
3/x^(1/2) = 6 / 7
and solve for x:
x = (3 / (6/7))^2 = (7 / 2)^2 = 49/4 = 12.25, which is not an option provided. This indicates there might be a mistake in the provided options or the setup of the question.