Contraction Mapping Question: rx(1 − x)

Question

I came across the following question:

• Let (R, | · |) be a metric space. Let r ≥ 0 and define the mapping L : R → R where L(x) = rx(1 − x).
• When is this a contraction mapping?

Let x,y in R. Thus, we have:

• |L(x) - L(y)| = |rx(1-x) - ry(1-y)| = |r|.|x(1-x) - y(1-y)| = r.|x - x² - y + y²| ≤ r(|x-y| + |y² - x²|) = r|x-y| + r|y² - x²|

Since |y² - x²| ≥ 0, I cannot conclude when this is a contraction mapping.

Can someone help me? Thanks in advance.