What is the connection between roots of f and g using Rolle's Theorem?

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In summary, using Rolle's Theorem, we can prove that between any two roots of f, there exists at least one root of g.
  • #1
Tomp
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[h=1][/h]I'm doing a question and I am getting stuck and need help

Question:
Consider the continuous functions f(x) = 1 - e^(x)*sin(x) and g(x) = 1 + e^(x)*cos(x). Using Rolle's Theorem, prove that between any two roots of f there exists at least one root of g.

Hint
Remember that, a root of f is a point x in the domain of f such that f(x) = 0.

Can someone provide a natural language proof of this?
 
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  • #2
Let a and b be two successive roots of f. Observe that by Rolle's theorem, we must have some c in (a,b) such that f'(c)=0. This means we must have:

$\displaystyle -e^c\cos(c)-e^c\sin(c)=0$

$\displaystyle -e^c\sin(c)=e^c\cos(c)$

$\displaystyle 1-e^c\sin(c)=1+e^c\cos(c)$

$\displaystyle f(c)=g(c)$

So, we find that f and g meet at the turning points of f. This means g must have at least one root between two successive turning points of f.
 

FAQ: What is the connection between roots of f and g using Rolle's Theorem?

What is Rolle's Theorem?

Rolle's Theorem is a mathematical theorem that states if a function is continuous on a closed interval, and differentiable on the open interval, with the same function value at the endpoints, then there exists at least one point within the interval where the derivative of the function is equal to zero.

What is the significance of Rolle's Theorem?

Rolle's Theorem is an important theorem in calculus as it provides a way to prove the existence of critical points for a function. This can be used to find the maximum and minimum values of a function, as well as proving the existence of solutions to equations.

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

While both theorems deal with the behavior of a function on a closed interval, the Mean Value Theorem requires the function to be continuous on the closed interval and differentiable on the open interval. In contrast, Rolle's Theorem only requires the function to be continuous on the closed interval and differentiable on the open interval with the same function value at the endpoints.

How is Rolle's Theorem applied in real-world situations?

Rolle's Theorem can be used in various real-world situations, such as in economics to determine the optimal production level for a company or in physics to find the velocity of an object at a specific point in time. It can also be used in engineering to find the maximum stress on a structure.

Can Rolle's Theorem be applied to all functions?

No, Rolle's Theorem can only be applied to functions that are continuous on a closed interval and differentiable on the open interval. If a function does not meet these criteria, then Rolle's Theorem cannot be used to find critical points or prove the existence of solutions.

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