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) = x⁴ᐟ⁵, [0, 1]

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Step 1: Recall the Mean Value Theorem (MVT), which 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).

Step 2: Check the continuity of f(x) = x^(4/5) on the interval [0, 1]. Since x^(4/5) is a root function, it is continuous on [0, 1] because the domain of x^(4/5) is all non-negative x.

Step 3: Check the differentiability of f(x) = x^(4/5) on the interval (0, 1). The derivative f'(x) = (4/5)x^(-1/5) is defined for all x > 0, so f is differentiable on (0, 1).

Step 4: Consider the endpoint x = 0. The derivative f'(x) = (4/5)x^(-1/5) is not defined at x = 0, which means f is not differentiable at the endpoint x = 0.

Step 5: Conclude that the function f(x) = x^(4/5) does not satisfy the hypotheses of the Mean Value Theorem on the interval [0, 1] because it is not differentiable at x = 0.

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주요 개념

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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) such that f'(c) equals the average rate of change over [a, b]. This theorem helps in understanding the behavior of functions and is crucial for verifying if a function meets its conditions.

Continuity

Continuity of a function on an interval means that the function has no breaks, jumps, or holes in that interval. For the Mean Value Theorem to apply, the function must be continuous on the closed interval [a, b]. This ensures that the function behaves predictably and smoothly across the entire interval, which is necessary for finding a point where the instantaneous rate of change matches the average rate of change.

Differentiability

Differentiability refers to the existence of a derivative at each point in an interval. For the Mean Value Theorem, the function must be differentiable on the open interval (a, b). Differentiability implies continuity, but not vice versa, and ensures that the function has a well-defined tangent at every point in the interval, which is essential for applying the theorem.

Checking the Mean Value TheoremWhich 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) = x⁴ᐟ⁵, [0, 1]