Find the number c that satisfies the conclusion of the Mean Value Theorem.
f(x)=x/(x+2)
[1,4]
f'(c)=(f(4)-f(1))/(4-1)
f(4)=4/6=2/3
f(1)=1/3
f'(c)=1/9
f'(x)=(x+2-x)/(x+2)^2
2/(c+2)^2=1/9
(c+2)^2=18
c+2=3*sqrt(2)
To find the number c that satisfies the conclusion of the Mean Value Theorem for the given function f(x) = x/(x+2) on the interval [1,4], we need to follow these steps:
Step 1: Verify that the function f(x) is continuous on the closed interval [1,4].
The function f(x) = x/(x+2) is a rational function, which is continuous on its domain. The function is defined for all real numbers except x = -2, so it is continuous on the interval [1,4].
Step 2: Find the derivative of the function f(x).
To apply the Mean Value Theorem, we need to find the derivative of f(x).
Using the quotient rule, the derivative of f(x) = x/(x+2) is:
f'(x) = [(x+2)(1) - (x)(1)] / (x+2)^2
Simplifying this expression, we get:
f'(x) = 2 / (x+2)^2
Step 3: Find the average rate of change of the function f(x) over the interval [1,4].
The average rate of change of a function over an interval is given by the difference in function values divided by the difference in x-values.
So, the average rate of change of f(x) over [1,4] is:
(f(4) - f(1)) / (4 - 1)
Plugging in the function values, we get:
(4/(4+2) - 1/(1+2)) / 3
(4/6 - 1/3) / 3
(2/3 - 1/3) / 3
1/3 / 3
1/9
Step 4: Apply the Mean Value Theorem.
According to the Mean Value Theorem, if a function is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), then there exists at least one number c in (a,b) such that f'(c) = (f(b) - f(a))/(b - a).
Here, a = 1, b = 4, and the average rate of change is 1/9.
Therefore, we need to find a number c in (1,4) such that f'(c) = 1/9.
Setting f'(c) equal to 1/9, we have:
2/(c+2)^2 = 1/9
Cross-multiplying and simplifying, we get:
9 = (c+2)^2
Taking the square root of both sides, we have:
±3 = c+2
Solving for c, we get two possible values:
c = -2 + 3 = 1
c = -2 - 3 = -5
So, the number c that satisfies the conclusion of the Mean Value Theorem for the given function f(x) = x/(x+2) on the interval [1,4] is c = 1 and c = -5.