Question

Verify that the function sitisfies the two hypotheses of the Mean Value Theorem on the given interval, then find all numbers c that satisfy the concluxion of the Mean Value Theorem.

(a) f(x)=2x² −3x+1,[0,2]
(b) f(x)=x³+x−1,[0,2]

Answer 1

To verify if a function satisfies the hypotheses of the Mean Value Theorem, we need to check two conditions: continuity and differentiability. For the function f(x) = 2x² - 3x + 1 on the interval [0, 2], it satisfies both conditions and the conclusion of the Mean Value Theorem is satisfied at c = 1. Similarly, for the function f(x) = x³ + x - 1 on the interval [0, 2], it also satisfies both conditions and the conclusion of the Mean Value Theorem is satisfied at c = √4/3.

Step-by-step explanation:

To verify if a function satisfies the hypotheses of the Mean Value Theorem, we need to check two conditions:

1. The function must be continuous on the closed interval [a, b].
2. The function must be differentiable on the open interval (a, b).

(a) For the function f(x) = 2x² - 3x + 1 on the interval [0, 2], we can see that it is a polynomial function, which is continuous and differentiable everywhere. Therefore, it satisfies both prerequisites of the Mean Value Theorem. To find the numbers c that satisfy the conclusion of the Mean Value Theorem, we can calculate the derivative of f(x) using the power rule: f'(x) = 4x - 3. Setting this derivative equal to the average rate of change of the function over the interval [0, 2], (f(2) - f(0)) / (2 - 0) = ((2² - 3(2) + 1) - (2(0)² - 3(0) + 1)) / (2 - 0), we get 1 = 4c - 3. Solving for c, we find that c = 1.

(b) For the function f(x) = x³ + x - 1 on the interval [0, 2], similar to part (a), this is also a polynomial function that is continuous and differentiable everywhere. Therefore, it satisfies the prerequisites of the Mean Value Theorem. Calculating the derivative of f(x), we find f'(x) = 3x² + 1. Setting f'(x) equal to the average rate of change of the function over the interval [0, 2], (f(2) - f(0)) / (2 - 0) = ((2³ + 2 - 1) - (0³ + 0 - 1)) / (2 - 0), we get 5 = 3c² + 1. Solving for c, we find that c = √4/3.