[Mean Value Theorem]
f(x)=-3x^3 - 4x^2 - 2x -3 on the closed interval [0,8].
Find the smallest value of c that satisfies the conclusion of the Mean Value Theorem for this function defined on the given interval.
I got 8 - sqrt(5696) / -18 = 3.748436059 but it's not right.
I redid my math and I think I did my calculations wrong. I got 8 +- sqrt(8000)/ -18 but it's still wrong.
Nevermind, I had an error in my calculations. It was sqrt(8128). I got the answer.
To find the smallest value of c that satisfies the conclusion of the Mean Value Theorem, we need to determine if a specific function value corresponds to the average rate of change of the function over the interval [0, 8].
The Mean Value Theorem states that if a function f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one value c in (a, b) such that:
f'(c) = (f(b) - f(a)) / (b - a)
In this case, the function f(x) = -3x^3 - 4x^2 - 2x - 3 is continuous on the closed interval [0, 8] and differentiable on the open interval (0, 8).
To find the smallest value of c, we need to find the derivative of f(x) and set it equal to the average rate of change of f(x) over the interval [0, 8].
First, let's find the derivative of f(x):
f'(x) = -9x^2 - 8x - 2
Next, let's calculate the average rate of change of f(x) over the interval [0, 8]:
(f(8) - f(0)) / (8 - 0) = (-3(8)^3 - 4(8)^2 - 2(8) - 3 - (-3(0)^3 - 4(0)^2 - 2(0) - 3)) / 8
Simplifying this expression gives:
(-2047 - 128 - 16 - 3 - (-3)) / 8 = -2194 / 8 = -274.25
Now, we set the derivative equal to -274.25 and solve for c:
-9c^2 - 8c - 2 = -274.25
Simplifying this equation gives:
-9c^2 - 8c + 272.25 = 0
Now, we can solve this quadratic equation using the quadratic formula:
c = (-(-8) ± √((-8)^2 - 4(-9)(272.25))) / (2(-9))
Simplifying further gives:
c = (8 ± √(64 + 9797.5)) / (-18)
Calculating the square root inside the square root:
c = (8 ± √(9850.5)) / (-18)
This is where our calculations differ from yours. The expression should actually be:
c = (8 ± √9850.5) / (-18)
Using a calculator, we find that:
c ≈ (8 ± 99.2462) / (-18)
So, the two possible values of c are:
c ≈ (8 + 99.2462) / (-18) ≈ -3.7159
c ≈ (8 - 99.2462) / (-18) ≈ 6.2435
Since we are looking for the smallest value of c, the answer is approximately -3.7159.