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Unformatted text preview: Version OWL/AAABA w DGQlOc —— Gilbert — (78759} i This print—out should have 8' questions.
Multipie—choice questions may continue on
the next coiurnn or page M ﬁnd all choices
before answering. IIIInn
Eln BACK 001 10.0 points Use Lagrange Multipliers to determine the
maximum vaiue of fan y) 3 m2 — 2:12 + 3
subject to the constraint 9135,29): m2+y2w1= 0 1. maximum m 3
2. maximum :2 2
3. maximum = 1
4. maximum m 8 5. maximum = 4c0r‘rect Explanation:
The extreme values of f subject to the con—
straint g = 0 occur at solutions of (Vfllm, y) m MWXIE: :4) Now (vf)($7 y) w (217: '4y> 3
while (Vexxayl = (233,21!)
Thus 2m :2 2M, may = 2A3}, and so we have that
2m()\~ 1) m 0, 2y()\+2) = 0. From the first equation, we know that either
mmOorAml. Now,if}\=1we havefrom
the second equation that y = 0, and also that 9($=O) = mgml = 0, ore: :2: ii. However, if a: m 0, then we have from the
second equation that A m ~2, and also that We) w y2_1= 0, Dry = a1.
Consequently, the critical points are
(i1, 0) and (0, at).
Since ‘ 
f(j:1, 0) 3 4 and f(0,:ti)= 1, we thus see that max value = e . keywords: 002 10.0 points A company manufacturing cars at plant A
in Detroit and plant B in St. Louis has found
that its total cost function is 0 a x2+2y2+2$y+90m+30y+10 when 3: cars are produced at piant A and y at
plant B. What production levei at plant A
should the company set to minimize the costs
of production when ﬁlling an order for 110
cars? 1. prod plant A 2: 50 cars
2. prod plant A m 80 cars correct 3. prod plant A = 60 cars Version 004/AAABA m DGQlOc _ Gilbert — (787‘59) ' 2 4. prod plant A m? 40 cars 5. prod plant A m 70 cars Explanation:
To determine the production level rnirnin—
imzing costs we have to minimize C m x2+2y2+2scy+90m+30y+1m a function of two variables, subject to the
restriction n+7; 2110. Using this last condition to eliminate y from
0 reduces the problem to optimization of a
function C(93) m 3:2 + 2(110 —— 51:)2 + 23:(110 w m)
+ 90:12 + some m as) + 10
= 3:2 ~ 160$ + 27510. of one variable. After differentiation, O’(m) = 21m 160, C”(;n) = 2 By the second derivative test for functions of
one variable 0 thus has a minimum value at
m = 80 t. a, to minimize production costs the
company should set the production level at plant/i m 80 cars . keywords: 003 10.0 points Locate and classify the iocal extremum of
' f when 1
= 3x+y+—+4, 3 my (333 y > G)‘ flat, 1;) 1. saddie at (3, 3) 2. saddie at (g, 3) 3. local min'at (3, 3) 1 4. locai max at (§’ 3) 5. local max at (3, 3) 1
6. local min at (3" 3) correct
Explanation: Differentiating once we see that At a local extremum these ﬁrst partiai deriva»
tives are zero. Thus f has a locai extremum
at (31 3) I To classify the local extremnm we use the ' second derivative test Now 2 1 2 film 3 "ﬁg: f$y_ — 3:23“;i fyy“ "' 333$
But then
A 2 fm (£3) = 18 > 0
2
C w fyy (#3) — 5 >. 0,
and
= x ‘m 1.
fy (an
1 Thus
140nt = 3 > 0. Hence by the second. derivative test3 f has a local min at (—331, 3) ...
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 Fall '09
 Gilbert

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