# Real analysis, simpl(er) questions

1. Apr 24, 2012

### rayred

1. The problem statement, all variables and given/known data

It is a 4 parter, but i got 3 and 4 done.

a) Find f ([0,3]) for the following function:
f(x)=1/3 x^3 − x + 1

b) Consider the following function :
f(x) = e^(−ax) (e raised to the power of '-a' times 'x') a, x ∈ [0,∞)
Find values of a for which f is a contraction .

3. The attempt at a solution
You would think that a is simple to me, but it is not. How does one go about solving a, because I have in my notes to take the max and min of [0, 3] then evaluate at those points, then do something with f prime? Im all confused because I must have screwed up my notes. Should not be too hard to answer

Now, I think I know what a contraction is, but I seem to be having problem
A contraction is something defined as such:

| f(x) - f(y) | <= n|x - y| for some 0 < n < 1... correct?
am i setting
f(x) = e^(-a_1x)
f(y) = e^(-a_2x)
so are we finding a value a that ensures 0 < n < 1 ? if so how?