Answer: C = 1/3
Explanation:
To determine if the function satisfies the hypotheses of the Mean Value Theorem on the given interval [1, 4], we need to check two conditions:
1. Continuity: The function must be continuous on the interval [1, 4].
2. Differentiability: The function must be differentiable on the open interval (1, 4).
Let's analyze the function f(x) = x/(x + 2):
1. Continuity: The function f(x) is continuous for all values of x except where the denominator equals zero. In this case, the denominator x + 2 equals zero when x = -2. However, the interval [1, 4] does not include -2, so the function is continuous on the interval [1, 4].
2. Differentiability: The function f(x) is differentiable for all values of x except where the denominator equals zero or where the function is discontinuous. In this case, the denominator x + 2 equals zero when x = -2. However, as mentioned earlier, the interval [1, 4] does not include -2. Therefore, the function is differentiable on the open interval (1, 4).
Since the function satisfies both the conditions of continuity and differentiability on the interval [1, 4], we can conclude that it satisfies the hypotheses of the Mean Value Theorem.
Now, to find all the numbers c that satisfy the conclusion of the Mean Value Theorem, we can use the following formula:
f'(c) = (f(b) - f(a))/(b - a)
where a and b are the endpoints of the interval [1, 4].
Calculating the values:
f'(c) = (f(4) - f(1))/(4 - 1)
f(4) = 4/(4 + 2) = 4/6 = 2/3
f(1) = 1/(1 + 2) = 1/3
f'(c) = (2/3 - 1/3)/(4 - 1) = 1/3
Therefore, the number c that satisfies the conclusion of the Mean Value Theorem is c = 1/3.