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) = sqrt(x) − (1/9)x, [0, 81]
c=?
f(0)=0
f(9)=0
So, that satisfies the theorem
f is continuous and differentiable.
So, we just have to find c such that f'(c)=0
f'(x) = 1/(2√x) - 1/9
1/(2√c) - 1/9 = 0
1/(2√c) = 1/9
2√c = 9
c = 81/4 which is in [0,81]
To verify that the function satisfies the three hypotheses of Rolle's Theorem, we need to check the following conditions:
1. Continuity: The function must be continuous on the closed interval [0, 81].
2. Differentiability: The function must be differentiable on the open interval (0, 81).
3. Equal function values: The function must have the same values at the endpoints of the interval, i.e., f(0) = f(81).
Let's check each condition:
1. Continuity: The function f(x) = sqrt(x) - (1/9)x is a combination of two continuous functions: sqrt(x) and -(1/9)x. Both of these functions are continuous for all nonnegative values of x. Therefore, f(x) is continuous on the closed interval [0, 81].
2. Differentiability: To check differentiability, we need to calculate the derivative of f(x). The derivative of f(x) = sqrt(x) - (1/9)x can be found by applying the power rule and the constant rule:
f'(x) = (1/2)sqrt(x) - (1/9)
The derivative is defined and continuous on the open interval (0, 81). Therefore, the function satisfies the differentiability condition.
3. Equal function values: We need to determine the values of f(0) and f(81) and check if they are equal.
f(0) = sqrt(0) - (1/9)(0) = 0
f(81) = sqrt(81) - (1/9)(81) = 9 - 9 = 0
Since f(0) = f(81) = 0, the function satisfies the equal function values condition.
Now that we have verified that the function satisfies all three hypotheses of Rolle's Theorem, we can find all the numbers c that satisfy the conclusion.
According to Rolle's Theorem, if a function satisfies the three hypotheses, then there exists at least one number c in the open interval (0, 81) such that f'(c) = 0.
To find the numbers c, we need to solve the equation f'(c) = 0:
(1/2)sqrt(c) - (1/9) = 0
Simplifying the equation, we have:
1/2 * sqrt(c) = 1/9
Squaring both sides, we get:
(c/4) = 1/81
Solving for c, we have:
c = 4/81 * 81
c = 4
Therefore, the number c that satisfies the conclusion of Rolle's Theorem is 4.
The answer is c = 4.
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) = sqrt(x) - (1/9)x is continuous on the closed interval [0, 81] because both the square root function and the linear function (1/9)x are continuous on their respective domains.
2. Differentiability: The function f(x) = sqrt(x) - (1/9)x is differentiable on the open interval (0, 81) because both the square root function and the linear function (1/9)x are differentiable on their respective domains.
3. Values at endpoints: We need to check if f(0) = f(81). Evaluating the function at the endpoints:
f(0) = sqrt(0) - (1/9)(0) = 0 - 0 = 0
f(81) = sqrt(81) - (1/9)(81) = 9 - 9 = 0
Since f(0) = f(81) = 0, the function satisfies this hypothesis as well.
Now, we can move on to finding the number(s) c that satisfy the conclusion of Rolle's theorem, which states that if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and has equal function values at the endpoints, there exists at least one number c in the open interval (a, b) such that f'(c) = 0.
Since f(x) = sqrt(x) - (1/9)x is differentiable on the open interval (0, 81), and f(0) = f(81) = 0, we can apply Rolle's theorem.
We need to find c such that f'(c) = 0. Taking the derivative of f(x):
f'(x) = (1/2)√(x) - (1/9)
Setting f'(x) = 0 and solving for x:
(1/2)√(x) - (1/9) = 0
(1/2)√(x) = (1/9)
√(x) = 2/9
x = (2/9)^2
x = 4/81
So, the number c that satisfies the conclusion of Rolle's theorem is c = 4/81.