Question

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) = {2x − 3, 0 ≤ x ≤ 26x − x² − 7, 2 \u003c x ≤ 3

Accepted Answer

Step 1: Understand the Mean Value Theorem (MVT). The 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). Step 2: Check continuity of f(x) on the given intervals. For the piecewise function f(x), check if it is continuous at the point where the pieces meet, which is x = 2. Evaluate the left-hand limit and the right-hand limit at x = 2 to ensure they are equal. Step 3: Check differentiability of f(x) on the open intervals. For each piece of the function, determine if the function is differentiable. For the first piece, f(x) = 2x - 3, check differentiability on (0, 2]. For the second piece, f(x) = 6x - x² - 7, check differentiability on (2, 3). Step 4: Verify the differentiability at x = 2. Since the function is piecewise, ensure that the derivative from the left and the derivative from the right at x = 2 are equal. Calculate the derivatives of each piece and compare them at x = 2. Step 5: Conclude whether the function satisfies the hypotheses of the MVT. If the function is both continuous on [0, 3] and differentiable on (0, 3), then it satisfies the MVT. Otherwise, identify which condition fails and explain why the MVT does not apply.

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) = {2x − 3, 0 ≤ x ≤ 2
6x − x² − 7, 2 < x ≤ 3

Risposta video verificata per un problema simile:

Concetti chiave

Mean Value Theorem

Continuity

Differentiability

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) = {2x − 3, 0 ≤ x ≤ 26x − x² − 7, 2 < x ≤ 3