Help with Rolle's theorem

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In summary, the conversation discusses a one-parameter family of planes defined by the equation x · n(u) + p(u) = 0, where n and p are vectors and u is a parameter. It is stated that two planes with different parameters intersect in a line, and this line also lies in the plane defined by x · (n(u_1) - n(u_2)) + p(u_1) - p(u_2) = 0. The conversation then mentions the use of Rolle's theorem, but there are questions about how it applies in this situation.
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monea83
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Given is a one-parameter family of planes, through

[tex]x \cdot n(u) + p(u) = 0[/tex]

with normal vector n and base point p, both depending on the parameter u.

Two planes with parameters [tex]u_1[/tex] and [tex]u_2[/tex], with [tex]u_1 < u_2[/tex], intersect in a line (planes are assumed to be non-parallel). This line also lies in the plane

[tex]x \cdot (n(u_1) - n(u_2)) + p(u_1) - p(u_2) = 0[/tex]

Now, the book I am reading claims that, "by Rolle's theorem, we get:"

[tex]x_1 n_1'(v_1) + x_2 n_2'(v_2) + x_3 n_3'(v_3) + p'(v_4) = 0[/tex] with [tex]u_1 \leq v_i \leq u_2[/tex].

However, I don't see how the theorem applies here... for starters, I don't see anything of the form [tex]f(a) = f(b)[/tex], as required by the theorem.
 
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monea83 said:
Given is a one-parameter family of planes, through

[tex]x \cdot n(u) + p(u) = 0[/tex]

with normal vector n and base point p, both depending on the parameter u.

I'm trying to understand your notation. If n and p are vectors, I supose x is a vector dotted into n, which gives a scalar?? How do you add a scalar to a vector? Is the 0 on the right side a scalar or vector? And you say you have a one parameter family of planes through

[tex]x \cdot n(u) + p(u) = 0[/tex]

What do you mean by that?
 
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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 and differentiable on the open interval, and the function's values at the endpoints of the interval are equal, then there exists at least one point within the interval where the derivative of the function is equal to zero.

2. Why is Rolle's theorem important?

Rolle's theorem is important because it provides a way to prove the existence of a critical point, or a point where the derivative of a function is equal to zero, within a given interval. This can be useful in many real-world applications, such as optimization problems and in physics and engineering.

3. How is Rolle's theorem related to the Mean Value Theorem?

Rolle's theorem is a special case of the Mean Value Theorem, which states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point within the interval where the derivative of the function is equal to the average rate of change of the function over the interval.

4. Can Rolle's theorem be applied to all types of functions?

Rolle's theorem can only be applied to differentiable functions, meaning that they have a well-defined derivative at every point within the interval. Additionally, the function must also be continuous on the closed interval.

5. How is Rolle's theorem used in real-world applications?

Rolle's theorem can be used to solve optimization problems, where the goal is to find the maximum or minimum value of a function within a given interval. It is also used in physics and engineering to determine critical points and to analyze the behavior of different systems.

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