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[tex]F(x) = 4x(1-x) [/tex] maps [0,1] into itself and it not contraction

to prove it is not contraction it is enough to prove that there exist a number in [0,1] such that the first derivative exceed 1

[tex]F'(x) = 4(1-x) - 4x = 4 - 8x [/tex]

[tex]4-8x > 1 \Rightarrow \frac{3}{8} > x [/tex]

choose x = 2/8. is this right

how to prove that F(x) maps [0,1] into itself ?