Find all numbers c that satisfy the conclusion of Rolle's Theorem for the following function.

If there are multiple values, separate them with commas; enter N if there are no such values.
f(x)= x^2-10x+3, [0,10]

Rolle's Theorem needs f(0) = f(10).

This is true.

So, we know that there is a c such that f'(c) = 0, where 0 <= c <= 10.

f'(x) = 2x-10
So, if f'(c) = 0, c=5.
c is indeed in the interval.

Well, well, well, look who we have here, Rolle's Theorem making an appearance! Let's give this a shot.

First things first, we need to check if our function, f(x) = x^2 - 10x + 3, satisfies the requirements of Rolle's Theorem. To do that, we need to make sure two things hold true: 1) f(x) is continuous on the closed interval [0,10], and 2) f(x) is differentiable on the open interval (0,10).

Now, f(x) = x^2 - 10x + 3 is a polynomial function, and we know that all polynomials are continuous and differentiable. So, we're good on both requirements!

But does this mean we have a value for c that satisfies the conclusion of Rolle's Theorem? Well, since f(x) is continuous and differentiable on [0,10], and the function is the same at the endpoints (f(0) = f(10)), we can conclude that there exists at least one value c in the open interval (0,10) such that f'(c) = 0.

Now, let's find that value of c. Let's take the derivative of f(x), which gives us f'(x) = 2x - 10. To find the critical points (where f'(x) = 0), we set f'(x) equal to zero and solve: 2x - 10 = 0. Solving this equation gets us x = 5.

So, the only value that satisfies the conclusion of Rolle's Theorem for the function f(x) = x^2 - 10x + 3 on the interval [0,10] is c = 5.

Hope that brings a smile to your face!

To apply Rolle's Theorem, we need to check the conditions of the theorem:

1. The function f(x) is continuous on the closed interval [0, 10].
2. The function f(x) is differentiable on the open interval (0, 10).
3. f(0) = f(10), meaning the function has the same value at the endpoints of the interval.

Let's check these conditions:

1. The function f(x) = x^2 - 10x + 3 is a polynomial, and polynomials are continuous for all real numbers. So, f(x) is continuous on [0, 10].

2. The derivative of f(x) = x^2 - 10x + 3 is f'(x) = 2x - 10. The derivative is a linear function, and linear functions are differentiable on all real numbers. So, f(x) is differentiable on (0, 10).

3. Now let's check if f(0) = f(10):
f(0) = (0)^2 - 10(0) + 3 = 3
f(10) = (10)^2 - 10(10) + 3 = 100 - 100 + 3 = 3

Since f(0) = f(10), all the conditions of Rolle's Theorem are satisfied.

According to Rolle's Theorem, there exists at least one value 'c' in the open interval (0, 10) where the derivative of f(x) is equal to zero.

To find this value 'c', we need to find the roots of the derivative f'(x) = 2x - 10:

2x - 10 = 0
2x = 10
x = 5

So, the value 'c' that satisfies the conclusion of Rolle's Theorem is c = 5.

Therefore, the only value that satisfies the conclusion of Rolle's Theorem for the function f(x) = x^2 - 10x + 3 on the interval [0, 10] is c = 5.

To apply Rolle's theorem, we need to check if the given function satisfies three conditions:

1. The function must be continuous on the closed interval [0, 10].
2. The function must be differentiable on the open interval (0, 10).
3. The function must have the same values at the endpoints, f(0) = f(10).

Let's check these conditions one by one.

1. Continuity: The function f(x) = x^2 - 10x + 3 is a quadratic function, and all quadratic functions are continuous on their entire domain, including the interval [0, 10]. So this condition is satisfied.

2. Differentiability: To check if f(x) is differentiable on the open interval (0, 10), we need to find its derivative. Taking the derivative of f(x):
f'(x) = 2x - 10.

This derivative is defined for all x-values, so f(x) is differentiable on the open interval (0, 10). Hence, this condition is also satisfied.

3. Equal values at endpoints: Now we need to check if f(0) = f(10). Let's calculate these values:
f(0) = (0)^2 - 10(0) + 3 = 0 - 0 + 3 = 3.
f(10) = (10)^2 - 10(10) + 3 = 100 - 100 + 3 = 3.

Since f(0) = f(10) = 3, this condition is satisfied as well.

Therefore, all the conditions of Rolle's theorem are satisfied for the function f(x) = x^2 - 10x + 3 on the interval [0, 10].

According to Rolle's theorem, there exists at least one number c in the open interval (0, 10) such that f'(c) = 0.

To find the values of c, we need to solve the equation f'(c) = 0.

f'(c) = 2c - 10 = 0.

Solving this equation, we find:
2c = 10,
c = 5.

So, the value of c that satisfies the conclusion of Rolle's theorem is c = 5.

Therefore, the only value that satisfies the conclusion of Rolle's theorem for the given function on the interval [0, 10] is c = 5.