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shrug
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1. How can you prove if F is continuous, then there exists a fixed point of F in [0,1]?
I know F:[0,1] ---> [0,1] an bijective, but what is f(c)=c mean?
I know F:[0,1] ---> [0,1] an bijective, but what is f(c)=c mean?
Dick said:I think it means more or less exactly what it says. There is a c in [0,1] such that F(c)=c. Use the intermediate value theorem. If F is bijective there is an a such that F(a)=0 and a b such that F(b)=1. What happens in between? Apply the IVT to F(x)-x on the interval [a,b].
The Intermediate Value Theorem is a mathematical concept that states that if a continuous function has a different sign at each end of an interval, then it must have at least one root (or zero) within that interval.
The Intermediate Value Theorem is used to prove the existence of solutions or roots in a given interval for a continuous function. This can be helpful in solving equations or problems where the exact solution cannot be found algebraically.
The Intermediate Value Theorem can only be applied to continuous functions. This means that the function must be defined and continuous over the entire interval in question.
No, the Intermediate Value Theorem can only prove the existence of a solution within an interval, but it does not provide the exact value of the solution. Additional methods, such as numerical approximations or algebraic manipulation, must be used to find the exact value.
Yes, the Intermediate Value Theorem can only be used to find a solution within a given interval. It does not provide any information about the number of solutions or their exact locations. Additionally, the function must be continuous for the theorem to be applied, which may not always be the case in real-world situations.