Verify that the function f(x)=x^3-6x^2+8x+4 satisfies the three hypotheses of Rolle's Theorem on the given interval [0,4]. Then find all numbers c that satisfy the conclusion of Rolle's Theorem
To verify that the function f(x) = x^3 - 6x^2 + 8x + 4 satisfies the three hypotheses of Rolle's Theorem on the interval [0, 4], we need to show the following:
1. f(x) is continuous on the closed interval [0, 4].
2. f(x) is differentiable on the open interval (0, 4).
3. f(0) = f(4).
Let's check each hypothesis step by step:
1. f(x) is continuous on the closed interval [0, 4]:
To show this, we need to check if f(x) is continuous for all values of x between 0 and 4, inclusive. The function f(x) = x^3 - 6x^2 + 8x + 4 is a polynomial, and polynomials are continuous for all real numbers. Thus, f(x) is continuous on [0, 4].
2. f(x) is differentiable on the open interval (0, 4):
To verify differentiability, we need to check if f(x) is differentiable for all values of x between 0 and 4, excluding the endpoints. The function f(x) = x^3 - 6x^2 + 8x + 4 is a polynomial, and polynomials are differentiable for all real numbers. Therefore, f(x) is differentiable on (0, 4).
3. f(0) = f(4):
To show whether f(0) is equal to f(4), we simply need to substitute x = 0 and x = 4 into the function f(x).
When x = 0:
f(0) = (0)^3 - 6(0)^2 + 8(0) + 4 = 4
When x = 4:
f(4) = (4)^3 - 6(4)^2 + 8(4) + 4 = 68
Since f(0) = 4 and f(4) = 68, we see that f(0) is not equal to f(4).
Based on our verification, the function f(x) = x^3 - 6x^2 + 8x + 4 satisfies the first two hypotheses of Rolle's Theorem. However, it does not satisfy the third hypothesis where f(0) is equal to f(4).
Therefore, we cannot apply Rolle's Theorem to find a number c that satisfies the conclusion of the theorem in this case.
To verify that the function f(x)=x^3-6x^2+8x+4 satisfies the three hypotheses of Rolle's Theorem on the interval [0,4], we need to check the following:
1. Continuity: f(x) is continuous on the closed interval [0,4].
2. Differentiability: f(x) is differentiable on the open interval (0,4).
3. f(0) = f(4): The function has the same value at the endpoints of the interval.
Let's check each hypothesis step by step:
1. Continuity: To determine the continuity of f(x), we need to check if there are any breaks or holes in the graph within the interval [0,4]. Since f(x) is a polynomial function, it is continuous for all real values of x. Therefore, f(x) is continuous on [0,4].
2. Differentiability: To determine the differentiability of f(x), we need to check if the derivative of f(x) exists for all x-values in the open interval (0,4). Taking the derivative of f(x), we get:
f'(x) = 3x^2 - 12x + 8.
The derivative exists for all real values of x, so f(x) is differentiable on (0,4).
3. f(0) = f(4): We need to calculate f(0) and f(4) and check if they are equal. Evaluating at x = 0 and x = 4:
f(0) = (0)^3 - 6(0)^2 + 8(0) + 4 = 4,
f(4) = (4)^3 - 6(4)^2 + 8(4) + 4 = 4.
Since f(0) = f(4) = 4, we can conclude that the third hypothesis is also satisfied.
Having determined that all three hypotheses of Rolle's Theorem are satisfied, we can now find the numbers c that satisfy the conclusion.
According to Rolle's Theorem, if a function satisfies the three hypotheses on the interval [a,b], then there exists at least one number c in the open interval (a,b) such that f'(c) = 0.
In our case, since the interval is [0,4], we need to find a number c in the open interval (0,4) such that f'(c) = 0. Taking the derivative f'(x), we already found:
f'(x) = 3x^2 - 12x + 8.
To find the values of c, we need to solve the equation f'(c) = 0:
3c^2 - 12c + 8 = 0.
Using factoring or the quadratic formula, we can solve for c. The solutions will be the values of c that satisfy the conclusion of Rolle's Theorem.