Final answer:
The mean value theorem states that for a function f(x) that is continuous on a closed interval and differentiable on an open interval, there exists at least one value c between the endpoints where the derivative is equal to the average rate of change of the function. In this case, the function is f(x) = √x - 2 and the interval is [2, 9]. By using the mean value theorem, we get c = 49/4.
Step-by-step explanation:
The mean value theorem states that for a function f(x) that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), there exists at least one value c between a and b such that the derivative of f(x) at c is equal to the average rate of change of f(x) over the interval [a, b].
In this case, the function is f(x) = √x - 2 and the interval is [2, 9].
To apply the mean value theorem, we need to check if the given function satisfies the conditions.
The function √x - 2 is continuous and differentiable for x > 2.
Therefore, we can use the mean value theorem to find the value of c.
First, let's find the derivative of f(x) = √x - 2.
The derivative is f'(x) = 1/(2√x).
Next, let's find the average rate of change of f(x) over the interval [2, 9].
The average rate of change is given by (f(9) - f(2))/(9 - 2).
Plugging in the values, we get (f(9) - f(2))/(9 - 2) = (√9 - 2 - (√2 - 2))/7 = (3 - 1)/7 = 2/7.
Now, we need to find the value of c between 2 and 9 such that f'(c) = 2/7.
Solving the equation 1/(2√c) = 2/7, we get c = 49/4.
Therefore, the mean value theorem guarantees that there exists a value c between 2 and 9 such that f'(c) = 2/7.