Fix α, a positive real number, and let X=[0,1] be a subset of R...?
Fix α, a positive real number, and let X=[0,1] be a subset of R. Define f:X->R by f(x)=αx(1-x) for x in X. For what values of α does the mapping f:X->R have the property that f(X) is a subset of X?
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The polynomial f(x) has two zeros, 0 and 1. The maximum value in that interval is at 0.5 (argue from symmetry or by taking the first derivative). f(0.5) = a*1/4, so a <= 4 will put an upper bound of 1 on f(x). a >= 0 puts a lower bound on f(x) of 0. Therefore if 0 <= a <= 4, f(X) will be a subset of X.
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