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) = 5 − 6x + 3x2, [0, 2]
can someone please explain this. thanks
f(0) = 5
f(2) = 5
so, that is ok.
f(x) is continuous
since f is a polynomial, it is differentiable
So, we want c such that f'(c) = 0
6c-6 = 0
c = 1
ok, since 1 is in [0,2].
To verify that the function satisfies the three hypotheses of Rolle's Theorem, we need to check the following:
1. Continuity: The function f(x) = 5 − 6x + 3x^2 is a polynomial, and polynomials are continuous on their domain, which in this case is the entire real numbers. So, the function is continuous on the closed interval [0, 2].
2. Differentiability: The function f(x) = 5 − 6x + 3x^2 is a polynomial, and all polynomials are differentiable everywhere. So, the function is differentiable on the open interval (0, 2).
3. Values at the endpoints: We need to check the function values at the endpoints of the interval [0, 2].
For x = 0:
f(0) = 5 - 6(0) + 3(0)^2 = 5
For x = 2:
f(2) = 5 - 6(2) + 3(2)^2 = 5 - 12 + 12 = 5
Since the function values at the endpoints are equal (5 = 5), the third hypothesis is satisfied.
Now, to find all numbers c that satisfy the conclusion of Rolle's Theorem, we need to find a point c in the open interval (0, 2) where the derivative of the function is zero.
First, let's find the derivative of f(x):
f'(x) = d/dx (5 − 6x + 3x^2) = -6 + 6x
Next, we'll set f'(x) = 0 and solve for x to find the critical point(s):
-6 + 6x = 0
6x = 6
x = 1
So, the critical point c that satisfies the conclusion of Rolle's Theorem is x = 1.
Therefore, the answer is c = 1.
To verify that a function satisfies the three hypotheses of Rolle's Theorem, we need to check the following:
1. Continuity: The function must be continuous on the closed interval [a, b].
2. Differentiability: The function must be differentiable on the open interval (a, b).
3. Endpoints: The function must have the same value at both endpoints, f(a) = f(b).
Let's go through each hypothesis for the given function f(x) = 5 - 6x + 3x^2 on the interval [0, 2].
1. Continuity:
To check if the function is continuous on [0, 2], we need to make sure that there are no breaks, jumps, or holes in the graph within the interval.
The given function f(x) is a polynomial function, and polynomial functions are continuous over their entire domain. Therefore, f(x) = 5 - 6x + 3x^2 is continuous on [0, 2].
2. Differentiability:
To check if the function is differentiable on (0, 2) (the open interval between 0 and 2), we need to ensure that the derivative of the function exists and is finite within the interval.
The derivative of f(x) is obtained by differentiating each term of the polynomial function:
f'(x) = -6 + 6x.
The derivative, f'(x), is a linear function which exists and is finite for all real numbers. Therefore, f(x) = 5 - 6x + 3x^2 is differentiable on (0, 2).
3. Endpoints:
To check if the function has the same value at both endpoints (f(0) = f(2)), we substitute x = 0 and x = 2 into the function and compare the results:
f(0) = 5 - 6(0) + 3(0)^2 = 5,
f(2) = 5 - 6(2) + 3(2)^2 = 5 - 12 + 12 = 5.
Since f(0) = f(2) = 5, the function f(x) = 5 - 6x + 3x^2 satisfies the endpoint condition.
Now that we have verified that all three hypotheses of Rolle's Theorem are satisfied, we can find the value(s) of c that satisfy the conclusion of Rolle's Theorem.
Rolle's Theorem states that if a function satisfies the three hypotheses, then there exists at least one number c in the open interval (a, b) such that f'(c) = 0.
The derivative of f(x) is f'(x) = -6 + 6x.
To find the critical point(s), we set f'(x) = 0 and solve for x:
-6 + 6x = 0
6x = 6
x = 1.
Therefore, the conclusion of Rolle's Theorem tells us that there exists at least one number c in the interval (0, 2) such that f'(c) = 0. In this case, the critical point x = 1 satisfies the conclusion.
So the number c that satisfies the conclusion of Rolle's Theorem for the given function f(x) = 5 - 6x + 3x^2 on the interval [0, 2] is c = 1.