Question

Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval? f(x) = х/x + 2' [1,4] Yes, it does not matter if f is continuous or differentiable; every function satisfies the Mean Value Theorem. O No, f is continuous on [1,4] but not differentiable on (1,4). Yes, f is continuous on [1, 4] and differentiable on (1,4). O No, f is not continuous on [1,4]. O There is not enough information to verify if this function satisfies the Mean Value Theorem. If it satisfies the hypotheses, find all numbers c that satisfy the conclusion of the Mean Value Theorem. (Enter your answers as a comma- separated list. If it does not satisfy the hypotheses, enter DNE). C=

Answer 1

Yes, the function satisfies the hypotheses of the Mean Value Theorem on the given interval. The number c that satisfies the conclusion of the Mean Value Theorem is c = -7/9.

Step-by-step explanation:

The function f(x) = x/x + 2 does satisfy the hypotheses of the Mean Value Theorem on the given interval [1, 4]. In order for a function to satisfy the Mean Value Theorem, it needs to be continuous on a closed interval and differentiable on the open interval. In this case, the function is continuous on [1, 4] and differentiable on (1, 4), so it satisfies the hypotheses of the Mean Value Theorem.

To find all numbers c that satisfy the conclusion of the Mean Value Theorem, we need to find the derivative of the function f(x) and then find the values of c for which the derivative is equal to the average rate of change of the function over the interval [1, 4].

The derivative of f(x) = x/x + 2 is f'(x) = 1/(x + 2). To find the average rate of change of the function over the interval [1, 4], we use the formula (f(b) - f(a))/(b - a), where a = 1 and b = 4. Plugging in these values, we get (f(4) - f(1))/(4 - 1) = (4/6 - 1/3)/(3) = (2/3 - 1/3)/(3) = 1/9.

To find the values of c that satisfy the conclusion of the Mean Value Theorem, we need to find the values of x for which f'(x) = 1/9. Solving the equation 1/(x + 2) = 1/9, we get x = -7/9. Therefore, the number c that satisfies the conclusion of the Mean Value Theorem is c = -7/9.