Find the number c that satisfies the conclusion of the Mean Value Theorem:
x^3+x-9 [0,2]
y = x^3 + x - 9
what is the change from 0 to 2 divided by 2?
y(2) = 8 + 2 - 9 = 1
y(0) = -9
change delta y = 1 -(-9) = 10
10/2 = 5 so somewhere between 0 and 2, dy/dx must be 5
dy/dx = 3 x^2 + 1
5 = 3 x^2 + 1
3 x^2 = 4
x^2 = 4/3
x = c = 1.15
To find the number c that satisfies the conclusion of the Mean Value Theorem for the function f(x) = x^3 + x - 9 over the interval [0, 2], we need to do the following:
1. First, let's find the average rate of change (slope) of the function f(x) over the interval [0, 2]. The average rate of change is given by the formula:
f'(c) = (f(b) - f(a)) / (b - a)
Where a and b are the endpoints of the interval, and f'(c) is the instantaneous rate of change at some point c within the interval.
In our case, a = 0, b = 2, and f(x) = x^3 + x - 9. Therefore, the average rate of change becomes:
f'(c) = (f(2) - f(0)) / (2 - 0)
2. Now, let's calculate f(2) and f(0). Inserting the values of x into the function f(x), we get:
f(2) = (2^3) + 2 - 9 = 8 + 2 - 9 = 1
f(0) = (0^3) + 0 - 9 = 0 + 0 - 9 = -9
3. Plug the values of f(2) and f(0) into the average rate of change formula:
f'(c) = (1 - (-9)) / (2 - 0) = 10 / 2 = 5
4. We have obtained the average rate of change f'(c) = 5. According to the conclusion of the Mean Value Theorem, there exists at least one value c in the interval [0, 2] such that the instantaneous rate of change (or derivative) at that point is equal to the average rate of change, which in our case is 5.
Therefore, the number c that satisfies the conclusion of the Mean Value Theorem is some value within the interval [0, 2] where the derivative of the function f(x) = x^3 + x - 9 is equal to 5. To find the exact value of c, we would need to take the derivative of f(x) and solve the equation f'(x) = 5.