Let $f_{c}$ be the map of the closed interval $[0,1]$ to itself given by

$$f_{c}(x)=c x(1-x), \text { where } 0 \leqslant c \leqslant 4 .$$

Sketch the graphs of $f_{c}$ and (without proof) of $f_{c}^{2}$, find their fixed points, and determine which of the fixed points of $f_{c}$ are attractors. Does your argument work for $c=3 ?$