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MATH 180\hfill FALL 1995

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%10:00 SECTION\hfill FONG

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\centerline{ASSIGNMENT \#32}

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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?

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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 $r=2R/3$.

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5.5, \#3.
$$S(x)={k_1\over x^2}+{k_2\over(20-x)^2}=k_2
  \left({7\over x^2}+{1\over(20-x)^2}\right)$$
when $k_1=7k_2$.  In particular,
$$ S'(x)=k_2\Bigl( {-14\over x^3}+{2\over
   (20-x)^3}\Bigr),\quad
   S''(x)=k_2\bigl( {42\over x^4}+{6\over(20-x)^4}\Bigr).$$
$S'(x)=0$ when $\displaystyle \Bigl ({x\over 20-x}\Bigr)^3=7$, that is,
when $\displaystyle \Bigl ({x\over 20-x}\Bigr) =\root 3 \of 7$.  This is
a linear equation in $x$ and has solution $x=13.13$.  Since
$S''(13.13)>0$, $S$ has a local minimum at $x=13.13$.

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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.

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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.

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