The function represents a parabola which crosses the x-axis at x=0 and x=1, and has its maximum value r/4 when x=1/2 (as you can verify by elementary calculus). So for , g(x) lies between 0 and r/4. It follows that if then g maps the unit interval into itself.

The fixed points of g are those for which g(x) = x, or . That is a quadratic equation, which you should be able to solve.