f(x) = x/(x+4), [1, 8]
To determine if 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 closed interval [1, 8] and differentiable on the open interval (1, 8).
First, we check for continuity:
The function f(x) = x/(x+4) is continuous on the closed interval [1, 8] because it is a rational function and is defined for all x in the interval.
Next, we check for differentiability:
The function f(x) = x/(x+4) is differentiable on the open interval (1, 8) because it is a rational function and the derivative exists for all x in the interval.
Since the function f(x) = x/(x+4) is both continuous on the closed interval [1, 8] and differentiable on the open interval (1, 8), it satisfies the hypotheses of the Mean Value Theorem.
To find all numbers c that satisfy the conclusion of the Mean Value Theorem, we can use the Mean Value Theorem formula:
f'(c) = (f(b) - f(a))/(b - a)
Where a = 1 and b = 8.
Now, we can find the derivative of f(x):
f'(x) = (1*(x+4) - x*1)/(x+4)^2
f'(x) = (x + 4 - x)/(x+4)^2
f'(x) = 4/(x+4)^2
Next, we can find f(8) and f(1):
f(8) = 8/(8+4) = 8/12 = 2/3
f(1) = 1/(1+4) = 1/5
Now, we can find f'(c) by setting f'(x) equal to the average rate of change:
4/(c+4)^2 = (2/3 - 1/5)/(8 - 1)
Solving for c will give us the numbers that satisfy the conclusion of the Mean Value Theorem.