MATH 180 FALL 1995

ASSIGNMENT #32

5.4, #3. C(0) is the fixed cost. Marginal cost the rate of increase of the cost. Since C(q) is increasing, marginal cost is positive. Marginal increase falls as q increases, but rises again after a certain point. Thus C(q) is concave-down on the first half of its domain, concave-up on the second half of its domain. The concavity changes where marginal cost has a local minimum. Are there goods with different types of cost function? The cost function takes the shape given in the problem for two reasons: As q increases, efficiency of scale lowers costs. But after a certain point, factors such as scarcity of supplies and labor raise costs. Is this so for Coca Cola?

5.5, #1. v = a(R−r)r^{2} = a(Rr^{2} −r^{3}), v′ = a(2Rr −3r^{2}) = ar(2R−3r^{2}), v′′ = a(2R−6r). Thus v has critical points at the roots r = 0 and r = 2R/3 of v. The second-derivative test implies r = 0 is a local minimum and r = 2R/3 is local maximum. In the problem, v is taken on the domain [0, R]. Both critical points are in [0, R] and v = 0 at both end points. Since v(2R/3) > 0, the maximal value of v on [0, R] is attained at r = 2R/3.

5.5, #3.

S(x) = \frac{k_{1}}{x^{2}} + \frac{k_{2}}{(20 − x)^{2}} = k_{2} (\frac{7}{x^{2}} + \frac{1}{(20 − x)^{2}} )

when k_{1} = 7k_{2}. In particular,

S′(x) = k_{2}(\frac{−14}{x^{3}} + \frac{2}{(20 − x)^{3}} ), S′′(x) = k_{2}(\frac{42}{x^{4}} + \frac{6}{(20 − x)^{4}} ).

S′(x) = 0 when (\frac{x}{20 − x})^{3} = 7, that is, when (\frac{x}{20 − x}) = ^{3} \sqrt{7}. This is a linear equation in

x and has solution x = 13.13. Since S′′(13.13) > 0, S has a local minimum at x = 13.13.

5.5, #4. s(x) = x − x^{2} has a critical point at x = 1/2 since s′(x) = 1 − 2x. This is clearly a local maximum since s(x) is a parabola opening downwards.

5.5, #5. f(x) = (x−a_{1})^{2} +(x−a_{2})^{2} +(x−a_{3})^{2}, f′(x) = 2(x−a_{1})+2(x−a_{2})+2(x−a_{3}) = 6x − 2(a_{1} + a_{2} + a_{3}). The critical point x = (a_{1} + a_{2} + a_{3})/3 is the arithmetic average of a_{1}, a_{2}, a_{3}, and is a local minimum by the first-derivative test.