Solving x^4 + 3x + c = 0 with Mean Value & Rolle's Theorems

In summary, the conversation discusses the concepts of Mean Value Theorem and Rolle's Theorem in relation to finding the number of roots of the equation x^4 + 3x + c = 0. It is mentioned that Rolle's Theorem is a corollary of the Mean Value Theorem and that it can be used to determine the number of roots in a given interval. The conversation also suggests applying Rolle's Theorem to intervals [a,b] and [b,c] to find a contradiction if the equation has more than two roots.
  • #1
powp
91
0
Hello

I am really not getting this Mean Value Theorem and Rolle's theorem.

I have this question

Show how that the equation x^4 + 3x + c = 0 has at most two roots.

How do I know what interval to use for Mean Value Theorem and Rolles theorem?? Think this is the part that confusses me. I think that different results can be obtained if I pick different intervals.

Please help this lost student

Peter
 
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  • #2
Rolle's Theorem is, as you should be able to see, just a corollary of the Mean Value Theorem. Since you're concerned specifically with roots, it's probably best to just look at Rolle's Theorem. Now suppose it has more than two roots. You know that this would mean that it has at least three. Let's call them a, b, and c with a < b < c. So what intervals would Rolle's Theorem apply to? What would it tell you about the derivative of some point in these intervals? What would this tell you about the number of roots f' has? Compute f' and figure out how many roots it actually has. You'll come to a contradiction.
 
  • #3
Suppose x^4 + 3x + c = 0 had 3 different roots: a< b< c. Apply Rolle's theorem to the intervals [a,b] and [b,c]. What does it tell you must be true about the derivative of x^4+ 3x+ c? What IS the derivative? Is it true?
 

FAQ: Solving x^4 + 3x + c = 0 with Mean Value & Rolle's Theorems

1. How do Mean Value and Rolle's Theorems help in solving equations?

Mean Value and Rolle's Theorems are useful mathematical tools that help in solving equations by providing a way to determine the existence and location of roots. Mean Value Theorem states that if a function is continuous on an interval and differentiable on the open interval, then at some point within the interval, the slope of the tangent line will be equal to the average slope of the function on that interval. Rolle's Theorem states that if a function is continuous on a closed interval and differentiable on the open interval, and if the function's values at the endpoints are equal, then there must be at least one point in the open interval where the derivative is equal to zero, indicating a potential root.

2. How do you apply Mean Value and Rolle's Theorems to solve x^4 + 3x + c = 0?

To apply Mean Value and Rolle's Theorems to solve x^4 + 3x + c = 0, we first need to rewrite the equation as a function f(x) = x^4 + 3x + c. Then, we can use the theorems to determine the existence and location of roots. First, we use Rolle's Theorem to find a potential root by setting f(a) = f(b) and solving for x. This will give us one potential root. Next, we can use the Mean Value Theorem to find other potential roots by setting the slope of the tangent line equal to the average slope of the function on the interval.

3. Can Mean Value and Rolle's Theorems be applied to any type of equation?

No, Mean Value and Rolle's Theorems are specifically used for solving equations that are continuous and differentiable on a given interval. If the function is not continuous or differentiable, then these theorems cannot be applied.

4. Are there any limitations to using Mean Value and Rolle's Theorems in solving equations?

Yes, there are a few limitations to using Mean Value and Rolle's Theorems. These theorems can only provide potential roots and cannot guarantee the exact location of the roots. Additionally, the equation must have at least one real root for these theorems to be applicable. If the equation has complex roots or no real roots, then these theorems cannot be used to solve it.

5. Can Mean Value and Rolle's Theorems be used to solve equations with multiple variables?

No, Mean Value and Rolle's Theorems are only applicable to functions with a single variable. They cannot be used to solve equations with multiple variables, as they require the function to be continuous and differentiable on a given interval, which is not possible for functions with multiple variables.

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