How do i find k in the banach fixed point theorem

[sara_87]

how do i find k in the banach fixed point theorem.
so say i have a function f(x)=1+3x-x^2 in the interval [1,2]
then how do i find k?

thank you

 

[jhicks]

It looks to be obvious that that function is a contraction, so apply the definition of a fixed point (k $$\in$$ [1,2] such that F(k) = k).

TeX is being weird so I don't know if what I said shows up properly.

 

[sara_87]

It looks to be obvious that that function is a contraction, so apply the definition of a fixed point (k $$\in$$ [1,2] such that F(k) = k).

TeX is being weird so I don't know if what I said shows up properly.

yes but how do i find that k?

 

[jhicks]

you know that f(k) = k. It's just algebra. What is f(k)?

 

[sara_87]

but what do i put as k?
what value of x isit?

 

[HallsofIvy]

k is what you are trying to find! F(k)= 1- 3k- k³ so F(k)= k is just 1- 3k- k²= k. Solve that equation!