verify that the function satisfies the hypothesis of the mean value theorem on the given interval. then find all numbers c that satisfy the conclusion of the mean value theorem.

f(x) = x/(x+2) , [1,4]

To verify that the function satisfies the hypothesis of the mean value theorem on the given interval [1, 4], we need to check two conditions:

1. Function Continuity: The function f(x) must be continuous on the closed interval [1, 4].
2. Function Differentiability: The function f(x) must be differentiable on the open interval (1, 4).

Let's start with the first condition:

1. Function Continuity:
To check if f(x) is continuous on the interval [1, 4], we need to ensure that the function is continuous at both endpoints (x = 1 and x = 4) and within the interval.

a. Endpoint 1: Setting x = 1, we have f(1) = 1 / (1 + 2) = 1/3. Since 1/3 is a finite value, the function is continuous at x = 1.

b. Endpoint 2: Setting x = 4, we have f(4) = 4 / (4 + 2) = 4/6 = 2/3. Again, 2/3 is a finite value, so the function is continuous at x = 4.

c. Interval (1, 4): The function f(x) = x / (x + 2) is a rational function and is continuous for all x ≠ -2.

Therefore, the function f(x) = x / (x + 2) is continuous on the closed interval [1, 4].

Next, let's check the second condition:

2. Function Differentiability:
To check if f(x) is differentiable on the open interval (1, 4), we need to ensure that the derivative of the function exists for all x within that interval.

The derivative of f(x) = x / (x + 2) can be found using the quotient rule:
f'(x) = [(x + 2)(1) - x(1)] / (x + 2)^2
= (2) / (x + 2)^2

Since the derivative f'(x) exists for all x ≠ -2 (except at x = -2, where there is a vertical asymptote), the function f(x) is differentiable on the open interval (1, 4).

Now that we have verified that the function satisfies the hypothesis of the mean value theorem, we can proceed to find all the numbers c that satisfy the conclusion of the mean value theorem.

According to the Mean Value Theorem, there exists at least one number c in the open interval (1, 4) such that:
f'(c) = [f(4) - f(1)] / (4 - 1)

Let's calculate the right-hand side of the equation:
[f(4) - f(1)] / (4 - 1) = [(2/3) - (1/3)] / 3
= (1/3) / 3
= 1/9

Now, we need to find the number c in the interval (1, 4) where f'(c) equals 1/9.

Using the derivative of f(x), we can set it equal to 1/9:
(2) / (c + 2)^2 = 1/9

To solve this equation, we can cross-multiply:
2 = (c + 2)^2 / 9
18 = (c + 2)^2

Taking the square root of both sides:
±√18 = c + 2

This results in two possible values for c:
1) c = √18 - 2 (approximately 2.2426 - 2 = 0.2426)
2) c = -√18 - 2 (approximately -2.2426 - 2 = -4.2426)

Therefore, the numbers c that satisfy the conclusion of the mean value theorem are approximately 0.2426 and -4.2426.