Question

Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval? f(x)= x/x+4 [1, 8] Select the correct choice below:

A. Yes, it does.
B. No, it does not. C. Yes, it does not matter if f is continuous or differentiable; every function satisfies the Mean Value Theorem. D. No, it does not matter if f is continuous or differentiable; no function satisfies the Mean Value Theorem.

Answer 1

The function f(x) = x/(x+4) satisfies the hypotheses of the Mean Value Theorem on the interval [1, 8].

Step-by-step explanation:

To determine whether the function satisfies the hypotheses of the Mean Value Theorem on the given interval, we need to check if the function is continuous on the interval and differentiable on the open interval.

1. To check for continuity, we need to ensure that the function is defined and has no breaks or jumps in the interval [1, 8].

• In this case, the function f(x) = x/(x+4) is defined for all values of x in the interval [1, 8].
• There are no vertical asymptotes or any other breaks or jumps in the function within the interval.
• Therefore, the function is continuous on the interval [1, 8].

To check for differentiability, we need to ensure that the derivative of the function exists and is finite on the open interval (1, 8).

• The derivative of f(x) = x/(x+4) is f'(x) = -4/(x+4)^2.
• The derivative exists and is finite for all values of x in the open interval (1, 8).
• Therefore, the function is differentiable on the open interval (1, 8).

Based on our analysis, the function f(x) = x/(x+4) satisfies the hypotheses of the Mean Value Theorem on the interval [1, 8].