Checking the Mean Value Theorem


Find the value or values of c that satisfy the equation (f(b) − f(a)) / (b − a) = f′(c) in the conclusion of the Mean Value Theorem for the functions and intervals in Exercises 1–6.


f(x) =√(x − 1), [1, 3]

Verified step by step guidance

1. First, ensure that the function f(x) = √(x - 1) is continuous on the closed interval [1, 3] and differentiable on the open interval (1, 3). Since the square root function is continuous and differentiable wherever its argument is positive, f(x) meets these conditions on the given interval.
2. Calculate f(a) and f(b) where a = 1 and b = 3. This involves evaluating the function at the endpoints of the interval: f(1) = √(1 - 1) = 0 and f(3) = √(3 - 1) = √2.
3. Apply the Mean Value Theorem formula: (f(b) - f(a)) / (b - a) = f'(c). Substitute the values: (√2 - 0) / (3 - 1) = f'(c). Simplify the left side to get √2 / 2.
4. Find the derivative f'(x) of the function f(x) = √(x - 1). Using the chain rule, f'(x) = 1/(2√(x - 1)).
5. Set the derivative equal to the simplified expression from the Mean Value Theorem: 1/(2√(c - 1)) = √2 / 2. Solve this equation for c to find the value(s) of c that satisfy the Mean Value Theorem.

Key Concepts

Mean Value Theorem

The Mean Value Theorem states that for a function f that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), there exists at least one point c in (a, b) such that f'(c) equals the average rate of change over [a, b]. This theorem connects the derivative of a function to its overall behavior on an interval.

Continuity and Differentiability

For the Mean Value Theorem to apply, the function must be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). Continuity ensures no breaks or jumps in the function, while differentiability ensures the function has a defined slope at every point within the interval. These conditions are crucial for finding the point c where the theorem holds.

Derivative Calculation

Calculating the derivative of the function f(x) = √(x − 1) is essential to apply the Mean Value Theorem. The derivative, f'(x), represents the instantaneous rate of change of the function. For f(x) = √(x − 1), using the chain rule, f'(x) = 1/(2√(x − 1)). This derivative helps find the specific value of c that satisfies the theorem's conclusion.