Yahoo Answers: Answers and Comments for Differential Calculus Optimization Problem? [Mathematics]
Copyright © Yahoo! Inc. All rights reserved.
https://ca.answers.yahoo.com/question/index?qid=20160325165250AAlAL3E
From Anonymous
enCA
Fri, 25 Mar 2016 16:52:50 +0000
3
Yahoo Answers: Answers and Comments for Differential Calculus Optimization Problem? [Mathematics]
292
38
https://ca.answers.yahoo.com/question/index?qid=20160325165250AAlAL3E
https://s.yimg.com/zz/combo?images/emaillogoca.png

From Thomas: f=xy+z(x+y)
pf/px=y+z, pf/py=x+z
The two par...
https://ca.answers.yahoo.com/question/index?qid=20160325165250AAlAL3E
https://ca.answers.yahoo.com/question/index?qid=20160325165250AAlAL3E
Fri, 25 Mar 2016 17:30:47 +0000
f=xy+z(x+y)
pf/px=y+z, pf/py=x+z
The two partial derivatives can only be equal when x=y....
since z=10xy we can say z=102x so the original equation can be expressed...
f=x^2+x(102x)+x(102x)
f=x^2+2x(102x)
f=x^2+20x4x^2
f=20x3x^2
df/dx=206x and since d2f/dx2=6, when df/dx=0 it is an absolute maximum for f(x)...
df/dx=0 only when 6x=20, x=10/3
(so x=y=10/3, z=10/3, then x=y=z=10/3)
Since f(x)=20x3x^2 the maximum value is:
f(10/3)=100/3
if you look back at the original equation...
3(10/3)^2=300/9=100/3

From az_lender: When one of the three numbers is nearly 0, the...
https://ca.answers.yahoo.com/question/index?qid=20160325165250AAlAL3E
https://ca.answers.yahoo.com/question/index?qid=20160325165250AAlAL3E
Fri, 25 Mar 2016 17:34:59 +0000
When one of the three numbers is nearly 0, the maximum is 25. When one of the three numbers is 1, the maximum is 20.25+4.5+4.5 = 29.25. When one of the three numbers is 2, the maximum is 16 + 8 + 8 = 32. When one of the three numbers is 3, the maximum is 12.25 + 10.5 + 10.5 = 33.25. When one of the three numbers is 10/3, the maximum is 3*(10/3)^2 = 33.3333... . When one of the three numbers is 4, the maximum is 9 + 12 + 12 = 33. It is plain that the maximum is 100/3, but I'll do something more like a proof:
The function representing the sum of the three products can be seen as a function of z only, viz.,
[(10z)/2]^2 + 2z(10z)
= 25  5z + z^2/4 + 20z  2z^2
= 25 + 15z  2.25 z^2.
The derivative of this function, with respect to z, is
15  4.5z, which is 0 when z = 15/4.5 = 10/3.
Then the max value is
(10/3)^2 + (10/3)^2 + (10/3)^2 = 100/3.