Intermediate Value Theorem and Rolle's Theorem to show root

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To demonstrate that the function f(x) = 2x - 2 - cos(x) has exactly one root, the Intermediate Value Theorem can be applied by evaluating f(0) and f(π), which show a sign change, indicating a root exists between these points. Additionally, Rolle's Theorem suggests that if there were two roots, there would be a point where the derivative is zero, contradicting the behavior of the function in the interval. The discussion emphasizes that "one root" means the graph intersects the x-axis only once. The combination of these theorems effectively confirms the uniqueness of the root. Thus, f(x) has exactly one root.
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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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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?
 

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