13. (a) The fixed points are the solutions of xn = s*xn2*(1 - xn), that is, xn = 0, (s - sqrt(s2 - 4*s))/2*s, and (s + sqrt(s2 - 4*s))/2*s. Note the second and third fixed points are real only for s <= 0 and s => 4.

(b) The slope of the tangent line is s*(2*xn - 3*xn2).

(c) At xn = 0, the slope of the tangent line is 0; at xn = 1, the slope of the tangent line is -s.

(d) The maximum occurs where the slope of the tangent line is zero, that is, at xn = 2/3.

(e) For xn = 2/3, xn+1 = 4*s/27. Thus xn+1 = 1 for s = 27/4 and so the dynamics are bounded between 0 and 1 for 0 <= s <= 27/4.

(f) The fixed point xn = 0 is stable for all s. The slope of the tangent line at xn = (s - sqrt(s2 - 4*s))/2*s is (6 - s + sqrt(s2 - 4*s))/2; this fixed point is never stable. Theslope of the tangent line at xn = (s + sqrt(s2 - 4*s))/2*s is (6 - s - sqrt(s2 - 4*s))/2; this fixed point is stable for 4 < s <= 16/3.

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