Please help me with this problem:

Find the number c that satisfies the conclusion of the Mean Value Theorem.
f(x) = x/(x + 4)
[1, 8]

i got to f'(x)= 4/(x+4)^2=(-1/60).

I do not see a number c defined in your f(x) equation, so don't know how to solve for it.

Is it the number between 1 and 8 for which f'(c) = [f(8) - f(1)]/(8-1) ?

The mean value theorem states that such a number c exists, as I recall.

To find the number c that satisfies the conclusion of the Mean Value Theorem, we need to follow these steps:

1. Find the derivative of the function f(x) = x/(x + 4). You correctly found the derivative as f'(x) = 4/(x + 4)^2.

2. Determine the interval over which the Mean Value Theorem is applicable. In this case, the given function f(x) is continuous and differentiable on the closed interval [a, b], where a = 1 and b = 8.

3. Calculate the average rate of change over the interval [a, b]. The average rate of change (denoted as f(b) - f(a)) is equal to the slope of the secant line connecting the points (a, f(a)) and (b, f(b)). Plugging in the values, we get:
f(b) - f(a) = f(8) - f(1)
= (8/(8 + 4)) - (1/(1 + 4))
= 8/12 - 1/5
= 4/6 - 1/5
= 2/3 - 1/5
= 10/15 - 3/15
= 7/15.

4. Calculate the derivative evaluated at some point c in the interval [a, b]. In this case, we need to solve for c in the equation f'(c) = (f(b) - f(a))/(b - a). Substituting the values, we get:
4/(c + 4)^2 = 7/15.

To find the value of c, we can solve the equation above for c. Here's how to do it:

First, multiply both sides of the equation by (c + 4)^2 to eliminate the denominator:
4 = (7/15)(c + 4)^2.

Next, multiply both sides by 15 to get rid of the fraction:
60 = 7(c + 4)^2.

Now, divide both sides by 7 to isolate (c + 4)^2:
(c + 4)^2 = 60/7.

Finally, take the square root of both sides to solve for c:
c + 4 = ±√(60/7).

Subtract 4 from both sides:
c = -4 ± √(60/7).

So, the number c that satisfies the conclusion of the Mean Value Theorem is:
c ≈ -4 + √(60/7) or c ≈ -4 - √(60/7).