Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. (Enter your answers as a comma-separated list.)
f(x) = x^3 − x^2 − 12x + 8, [0, 4]
f is a polynomial, so it is continuous and differentiable
f(4) = f(0) = 8, so that's ok
So, you want c where f'(c) = 0
That means you need
3c^2 - 2c - 12 = 0
c = 2.361
The other value is not in [0,4]
To verify that the function satisfies the three hypotheses of Rolle's Theorem, we need to check the following:
1. The function f(x) is continuous on the closed interval [0, 4].
2. The function f(x) is differentiable on the open interval (0, 4).
3. The values of f(0) and f(4) are equal.
1. The function f(x) = x^3 − x^2 − 12x + 8 is a polynomial and is therefore continuous on the entire real number line, including the closed interval [0, 4]. So, the first hypothesis is satisfied.
2. To check if a function is differentiable on an open interval, we need to check if its derivative exists at every point in that interval. The derivative of f(x) is f'(x) = 3x^2 - 2x - 12, which exists for all real numbers. Therefore, the function f(x) is differentiable on the open interval (0, 4). So, the second hypothesis is also satisfied.
3. To check if f(0) = f(4), we substitute x = 0 and x = 4 into the function f(x) and compare the values:
f(0) = (0)^3 − (0)^2 − 12(0) + 8 = 8
f(4) = (4)^3 − (4)^2 − 12(4) + 8 = 8
Since f(0) = f(4) = 8, the third hypothesis is satisfied.
Since all three hypotheses of Rolle's Theorem are satisfied, we can now find the numbers c that satisfy the conclusion of Rolle's Theorem. According to the theorem, there must exist at least one point c in the open interval (0, 4) such that f'(c) = 0.
To find the values of c, we need to find the values of x for which f'(x) = 0. The derivative of f(x) is f'(x) = 3x^2 - 2x - 12. Setting f'(x) = 0, we have:
3x^2 - 2x - 12 = 0
We can solve this quadratic equation for x using factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a),
where a = 3, b = -2, and c = -12.
Using the formula, we find:
x = (-(-2) ± √((-2)^2 - 4(3)(-12)))/(2(3))
= (2 ± √(4 + 144))/(6)
= (2 ± √148)/(6)
= (2 ± 2√37)/(6)
= (1 ± √37)/(3).
Therefore, the numbers c that satisfy the conclusion of Rolle's Theorem are (1 + √37)/3 and (1 - √37)/3.