Banach fixed point theorem application for f(x)=3πx-cos(pi*x/2)

Question

I feel extremly stupid right now.

So we did Banach fixed point theorem and i thought i got it.

Basically if an intervall maps onto itself and is monotonous and contracts then you have a unique fixed point on that interval. Now we found an excercise on the internet that does neither and still we are supposed to prove the existence of a fixed point via Banach.

The function in question is this function and the interval is [0,1]. the iteration would be g(x)=f(x)+x

So i tried but neither of the edges map onto the interval [0,1] with g(0)=-1 and g(1)=3π+1

and the contraction via derivative is a total wash since the derivative at no point drops under 6.

Im starting to feel like i completely misunderstood something

Thank you

Answer 1

Hint: the inverse function is a contraction.