Intermediate Value Theorem and Rolle's Theorem to show root

In summary, using the Intermediate Value Theorem and Rolle's Theorem, it can be shown that f(x) = 2x-2-cosx has exactly one root. This can be proven by checking the values of f(0) and f(pi), and noting that "one root" means the graph touches the x-axis only once. Rolle's theorem also suggests that if one root is found, there cannot be another root.
  • #1
Wessssss
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Homework Statement


Use the Intermediate Value Theorem and Rolle's Theorem to show that f(x) = 2x-2-cosx has exactly one root.


Homework Equations





The Attempt at a Solution


I'm not really sure what the question is asking for. the theorems I believe are to prove the existence of a point between a closed interval, but I have no interval, and what does it mean by "one root"
 
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  • #2
Wessssss said:

Homework Statement


Use the Intermediate Value Theorem and Rolle's Theorem to show that f(x) = 2x-2-cosx has exactly one root.

The Attempt at a Solution


I'm not really sure what the question is asking for. the theorems I believe are to prove the existence of a point between a closed interval, but I have no interval, and what does it mean by "one root"

Hint: Check f(0) and f(pi). And "one root" means the graph touches the x-axis only once. And if you find one root, what can you conclude from Rolle's theorem if you have another root?
 

1. What is the Intermediate Value Theorem?

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b], and takes on two values, f(a) and f(b), then it must also take on every value in between f(a) and f(b). In other words, if a curve starts and ends at different points, it must pass through every point in between.

2. How is the Intermediate Value Theorem used to show a root?

The Intermediate Value Theorem can be used to prove that a root (or solution) exists for a given function. If a function is continuous on a closed interval [a, b], and f(a) and f(b) have opposite signs, then there must be at least one point c between a and b where f(c) = 0. This means that the function crosses the x-axis at least once, or in other words, has at least one root on the interval.

3. What is Rolle's 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 [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in (a, b) where the derivative of the function is equal to 0. In other words, the slope of the tangent line at that point is 0.

4. How is Rolle's Theorem used to show a root?

Rolle's Theorem can be used to prove that a root exists for a given function. If a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), and f(a) = f(b), then there must be at least one point c in (a, b) where the derivative of the function is equal to 0. This means that the function has a horizontal tangent line at that point, which can only occur if the function crosses the x-axis at least once and has a root on the interval.

5. Can the Intermediate Value Theorem and Rolle's Theorem be used to find the exact value of a root?

No, the Intermediate Value Theorem and Rolle's Theorem only guarantee the existence of a root, they do not provide an exact value for the root. Further analysis and techniques, such as the bisection method or Newton's method, are needed to find the exact value of a root.

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