Checking the Mean Value Theorem


Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.


f(x) = {sinx / x, −π ≤ x < 0
0, x = 0

Verified step by step guidance

1. The Mean Value Theorem (MVT) 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 c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).
2. First, check the continuity of the function f(x) = sin(x)/x for x in the interval [-π, 0). The function is continuous for all x ≠ 0, but we need to consider the behavior as x approaches 0.
3. To determine the continuity at x = 0, consider the limit: lim(x→0) sin(x)/x. This limit is a standard result and equals 1, which suggests that the function can be extended to be continuous at x = 0 by defining f(0) = 1.
4. Next, check differentiability. The function sin(x)/x is differentiable for x ≠ 0. However, we need to consider the differentiability at x = 0. Since the function is not defined at x = 0 in the original form, we cannot directly apply the MVT.
5. Since the function is not differentiable at x = 0, it does not satisfy the hypotheses of the Mean Value Theorem on the interval [-π, 0].

Key Concepts

Mean Value Theorem

The Mean Value Theorem (MVT) states that for a function f that is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), there exists at least one point c in (a, b) where the derivative f'(c) equals the average rate of change over [a, b]. This theorem helps in understanding the behavior of functions and is crucial for determining if a function meets its conditions.

Continuity

Continuity of a function at a point means that the function's value approaches the same number from both sides of that point. For the Mean Value Theorem to apply, the function must be continuous on the entire closed interval [a, b]. In this context, checking continuity involves ensuring that the function does not have any breaks, jumps, or points of discontinuity within the interval.

Differentiability

Differentiability refers to the existence of a derivative at each point in an interval. A function is differentiable on an open interval (a, b) if it has a derivative at every point within that interval. For the Mean Value Theorem, differentiability is required on the open interval, meaning the function should be smooth without any sharp corners or cusps.