Fix a positive real number, let and define by

for

a. For what values of does the mapping have the property that ?

b. For what values of does the mapping have the property that and is a contraction?

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a. Since , then the derivative is

Since , I need to find the maximum value of the function to be 1, since to be a subset.

Therefore,

Plug x = 1/2 back into,

Thus, since we are given prior that alpha is a positive real number then where

b. To be a contraction, then its Lipschitz constant C for the mapping is

Then

So

I am having troubles with this inequality. Since , then and , how would I solve for the value of in ?

Thank you for your time.