Solve Rolle's Theorem for f on [1,3]

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In summary: Sqrt(3))/3 or (6+sqrt(3))/3In summary, the function f(x) = (x-1)(x-2)(x-3) satisfies the conditions for Rolle's theorem on the closed interval [1,3] as it is continuous and differentiable on (1,3). The derivative of the function is 3x^2 - 12x + 11 and to find the points in the interval with a slope of 0, we can use the quadratic formula to get c values of (6 - sqrt 3) / 3 and (6 + sqrt 3) / 3.
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Homework Statement


determine whether rolle's theorem can be applied to f on the closed interval [a,b]. If Rolle's theorem can be applied, find all values of c in the open interval (a,b) such that f'(c) = 0


Homework Equations


f(x) = (x-1)(x-2)(x-3) with the interval being [1,3]

The Attempt at a Solution


the function is continuous on [1,3] and differentiable on (1,3)

f(1) = 0 and f(3) = 0 so f(1) = f(3)

f'(x) = 3x^2 - 12x + 11

now I need to find the point in [1,3] with slope 0 so I set f'(x) = 0.

I know from here I'm supposed to get c values of (6 - sqrt 3) / 3 and (6 + sqrt 3) / 3 however the algebra to get me to this point is eluding me.

Can someone please point of the obvious for me?

Thanks
 
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  • #2
It is simply a quadratic equation:
[tex]3x^2 - 12x + 11 = 0[/tex]
You can solve it using your standard toolbox, e.g. by applying the quadrature formula:
[tex]x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/tex].
 
  • #3
use use the Quadratic equation to solve for x



x = (-b +/- sqrt(b^2 - 4ac)) / 2a


your equation is f'(x) 3x^2 - 12x + 11

Set for zero

3x^2 - 12x + 11=0


So.

a = 3 b = -12 c= 11

x = (12+/- Sqrt(144 - (4*3*11)))/6

x = (12 +/- Sqrt(12))/6

x = (Sqrt(3)+6)/3 or -(Sqrt(3)-6)/3
 

1. What is Rolle's Theorem?

Rolle's Theorem is a mathematical theorem that states that if a function is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), then there must be at least one point c in the interval (a,b) where the derivative of the function is equal to 0.

2. How do you apply Rolle's Theorem to a function?

To apply Rolle's Theorem, you must first check if the function is continuous on the closed interval [a,b] and differentiable on the open interval (a,b). If these conditions are met, then you can find the derivative of the function and set it equal to 0. This will give you the value of c, which is the point where the derivative is equal to 0 and satisfies the conditions of Rolle's Theorem.

3. What is the purpose of using Rolle's Theorem?

Rolle's Theorem is used to prove the existence of a point on a function where the derivative is equal to 0. This can be helpful in finding the maximum and minimum points of a function, as well as determining the behavior of a function on a given interval.

4. Can Rolle's Theorem be applied to any type of function?

No, Rolle's Theorem can only be applied to continuous functions on a closed interval [a,b] and differentiable on the open interval (a,b). If a function does not meet these conditions, then Rolle's Theorem cannot be applied.

5. What is the difference between Rolle's Theorem and the Mean Value Theorem?

Rolle's Theorem is a special case of the Mean Value Theorem. While Rolle's Theorem states that there exists at least one point where the derivative is equal to 0, the Mean Value Theorem states that there exists at least one point where the derivative is equal to the average rate of change of the function on a given interval. Additionally, Rolle's Theorem only applies to functions that have the same value at the endpoints of the interval, while the Mean Value Theorem does not have this restriction.

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