Solutions manual for the mathematics of optimization 4th edition 2021 quiz and answers/

CHAPTER 2 w




THE MATHEMATICS OF OPTIMIZATION
w w w




Thewproblemswinwthiswchapterwarewprimarilywmathematical.w Theywarewintendedwtowgivewstudentswsomewpracti
cewwithwtakingwderivativeswandwusingwthewLagrangianwtechniques,wbutwthewproblemswinwthemselveswofferwfe
wweconomicwinsights.wConsequently,wnowcommentarywiswprovided.wAllwofwthewproblemswarewrelativelywsim
plewandwinstructorswmightwchoosewfromwamongwthemwonwthewbasiswofwhowwtheywwishwtowapproachwthewteach
ingwofwthewoptimizationwmethodswinwclass.


Solutions

2.1 U w(x,wy)w=w4x2w +w3y 2
wUw
a. =w6wy
wUw w y
=w8xw
,
wx
b. 8,w12
w w
c. dUw =w dxw+w dyw=w8xw dxw+w6wyw dy
U U
wx wy
dyw
d. forwdUw =w0 8xwdxw+w6wyw dyw=w0
dx
dyw −8xw −4x
=w =w
dx 6wy 3y
e. xw=1, yw=w2 Uw =w4w1+w3w4w=16
dyw −4(1)w
f. =w =w−w2w/w3
dx 3(2)
g. Uw=w16wcontourwlinewiswanwellipsewcenteredwatwtheworigin.w Withwequation

4x2w+w3y 2 w =w16w,wslopewofwthewlinewatw(x,wy)w dyw =− w4xw .
is
dx 3y


2.2 a. Profitswarewgivenw byw w =w Rw−wCw =w−2q2w +w40qw−100
dw
=w−w4qw+w4 q*w=10
0
dq
dww*2w=w w−w2(10)2w+w40(10)w−100w=100
=− w
b. 2 4w sowprofitswarewmaximized
dq
dRw
c. MRw=w =w70w−w2
qwdq

,1

,
, 2w ❖w SolutionswManual


dCw
MCw=w =w2qw+w30
dq
sowq*w=w10wobeyswMRw=wMCw=w50.

2.3 Substitution:wyw=w1−w soww =wxyw=wxw−wx2
x f
fw
=w1w−w2wxw=w0
x
x=w0.5,wy= w0.5,w fw =w0.25

Note: fww=w−2ww0w.w Thiswiswawlocalwandwglobalwmaximum.
LagrangianwMethod:w ?w =wxyw+ww 1−wxw−wy)

£w
=w yw−ww=w0
x

£w
=wxw−ww=w0
y

so,wxw=wy.
usingwthewconstraintwgive xw=wyw=w0.5 xyw=w0.25
s ,

2.4 SettingwupwthewLagrangian:w ?w =wxw+wyw+ww 0.25w−wxy)w.
£w
=w1−w y
x
£w
=w1−w x
y
So,wxw=wy.w Usingwthewconstraintwgive xyw=wx2w =w0.25 xw=w yw=w0.5w.
s ,

2.5 a. fw(t)w=w−0.5gt2w+w40t

dfw 40w
=w−wgwtw+w40w=w0, t*w =w .
dt g

b. Substitutingwforwt* fw(t*)w=w−0.5g(4 g)2w+w40(40w g)w=w800w gw.
, 0
fw(t*)w
=− 2

800w gw .
g
c. f 1w *
g =− w (t ) dependswonwgwbecausewt wdependswonwg.
*

2

2
f =w−w0.5(t*)2w =w−0.5(w40)2w =w −800w .
sow