Calculus 3: Lagrange Multipliers
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Find the minimum and maximum of $f(x,y,z) = x^2+y^2+z^2$ subject to two constraints, $x+2y+z=8$ and $x−y=5$.
Looking at the equation, it's clear that there is no maximum.
After working this problem out, I found:
$x = 41/11$ , $y = -14/11$ , and $z = 9/11$
After plugging this into the original equation, I found the minimum to be $178/11$
However, my online homework is saying my answer is incorrect. Did I do something wrong?
Thank you in advance to anyone who can help me out with this.
calculus lagrange-multiplier constraints
edited Nov 13 at 6:20
Andrei
10.5k21025
asked Nov 13 at 5:07
EKM
262
add a comment |
up vote
1
down vote
favorite
Find the minimum and maximum of $f(x,y,z) = x^2+y^2+z^2$ subject to two constraints, $x+2y+z=8$ and $x−y=5$.
Looking at the equation, it's clear that there is no maximum.
After working this problem out, I found:
$x = 41/11$ , $y = -14/11$ , and $z = 9/11$
After plugging this into the original equation, I found the minimum to be $178/11$
However, my online homework is saying my answer is incorrect. Did I do something wrong?
Thank you in advance to anyone who can help me out with this.
calculus lagrange-multiplier constraints
edited Nov 13 at 6:20
Andrei
10.5k21025
asked Nov 13 at 5:07
EKM
262
2
To check whether or not you did something wrong, one would need to know what exactly you did.
– Dionel Jaime
Nov 13 at 5:11
The two constraints define a line. What are its min/max distances from the origin? In other words, check your work by solving the problem a different way.
– amd
Nov 13 at 6:26
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Find the minimum and maximum of $f(x,y,z) = x^2+y^2+z^2$ subject to two constraints, $x+2y+z=8$ and $x−y=5$.
Looking at the equation, it's clear that there is no maximum.
After working this problem out, I found:
$x = 41/11$ , $y = -14/11$ , and $z = 9/11$
After plugging this into the original equation, I found the minimum to be $178/11$
However, my online homework is saying my answer is incorrect. Did I do something wrong?
Thank you in advance to anyone who can help me out with this.
calculus lagrange-multiplier constraints
edited Nov 13 at 6:20
Andrei
10.5k21025
asked Nov 13 at 5:07
EKM
262
Find the minimum and maximum of $f(x,y,z) = x^2+y^2+z^2$ subject to two constraints, $x+2y+z=8$ and $x−y=5$.
Looking at the equation, it's clear that there is no maximum.
After working this problem out, I found:
$x = 41/11$ , $y = -14/11$ , and $z = 9/11$
After plugging this into the original equation, I found the minimum to be $178/11$
However, my online homework is saying my answer is incorrect. Did I do something wrong?
Thank you in advance to anyone who can help me out with this.
calculus lagrange-multiplier constraints
calculus lagrange-multiplier constraints
edited Nov 13 at 6:20
Andrei
10.5k21025
asked Nov 13 at 5:07
EKM
262
edited Nov 13 at 6:20
Andrei
10.5k21025
asked Nov 13 at 5:07
EKM
262
edited Nov 13 at 6:20
Andrei
10.5k21025
edited Nov 13 at 6:20
Andrei
10.5k21025
edited Nov 13 at 6:20
Andrei
10.5k21025
10.5k21025
asked Nov 13 at 5:07
EKM
262
asked Nov 13 at 5:07
EKM
262
asked Nov 13 at 5:07
EKM
262
262
2
To check whether or not you did something wrong, one would need to know what exactly you did.
– Dionel Jaime
Nov 13 at 5:11
The two constraints define a line. What are its min/max distances from the origin? In other words, check your work by solving the problem a different way.
– amd
Nov 13 at 6:26
add a comment |
2
To check whether or not you did something wrong, one would need to know what exactly you did.
– Dionel Jaime
Nov 13 at 5:11
The two constraints define a line. What are its min/max distances from the origin? In other words, check your work by solving the problem a different way.
– amd
Nov 13 at 6:26
2
2
To check whether or not you did something wrong, one would need to know what exactly you did.
– Dionel Jaime
Nov 13 at 5:11
To check whether or not you did something wrong, one would need to know what exactly you did.
– Dionel Jaime
Nov 13 at 5:11
The two constraints define a line. What are its min/max distances from the origin? In other words, check your work by solving the problem a different way.
– amd
Nov 13 at 6:26
The two constraints define a line. What are its min/max distances from the origin? In other words, check your work by solving the problem a different way.
– amd
Nov 13 at 6:26
add a comment |
2 Answers
2
active
oldest
votes
up vote
0
down vote
To check the result use that from the constraints
$y=x-5$
$z=8-x-2y=18-3x$
then we need to find the extrema for
$$g(x)=f(x,x-5,18-3x)=x^2+(x-5)^2+(18-3x)^2=11x^2-118x+349$$
answered Nov 13 at 6:17
gimusi
91.4k74495
add a comment |
up vote
0
down vote
I solved equation, and got an answer of $x=dfrac{23}{3}$ ,$y=dfrac{8}{3}$,$z=-5$. Is it the correct answer, by your online homework? If it is, I will share my solution with you. By the way, your answer doesn't fit the first constraint, you can check it by plugging in values.
answered Nov 21 at 11:46
Farid Hasanov
13
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
To check the result use that from the constraints
$y=x-5$
$z=8-x-2y=18-3x$
then we need to find the extrema for
$$g(x)=f(x,x-5,18-3x)=x^2+(x-5)^2+(18-3x)^2=11x^2-118x+349$$
answered Nov 13 at 6:17
gimusi
91.4k74495
add a comment |
up vote
0
down vote
To check the result use that from the constraints
$y=x-5$
$z=8-x-2y=18-3x$
then we need to find the extrema for
$$g(x)=f(x,x-5,18-3x)=x^2+(x-5)^2+(18-3x)^2=11x^2-118x+349$$
answered Nov 13 at 6:17
gimusi
91.4k74495
add a comment |
up vote
0
down vote
up vote
0
down vote
To check the result use that from the constraints
$y=x-5$
$z=8-x-2y=18-3x$
then we need to find the extrema for
$$g(x)=f(x,x-5,18-3x)=x^2+(x-5)^2+(18-3x)^2=11x^2-118x+349$$
answered Nov 13 at 6:17
gimusi
91.4k74495
To check the result use that from the constraints
$y=x-5$
$z=8-x-2y=18-3x$
then we need to find the extrema for
$$g(x)=f(x,x-5,18-3x)=x^2+(x-5)^2+(18-3x)^2=11x^2-118x+349$$
answered Nov 13 at 6:17
gimusi
91.4k74495
answered Nov 13 at 6:17
gimusi
91.4k74495
answered Nov 13 at 6:17
gimusi
91.4k74495
answered Nov 13 at 6:17
gimusi
91.4k74495
91.4k74495
add a comment |
add a comment |
up vote
0
down vote
I solved equation, and got an answer of $x=dfrac{23}{3}$ ,$y=dfrac{8}{3}$,$z=-5$. Is it the correct answer, by your online homework? If it is, I will share my solution with you. By the way, your answer doesn't fit the first constraint, you can check it by plugging in values.
answered Nov 21 at 11:46
Farid Hasanov
13
add a comment |
up vote
0
down vote
I solved equation, and got an answer of $x=dfrac{23}{3}$ ,$y=dfrac{8}{3}$,$z=-5$. Is it the correct answer, by your online homework? If it is, I will share my solution with you. By the way, your answer doesn't fit the first constraint, you can check it by plugging in values.
answered Nov 21 at 11:46
Farid Hasanov
13
add a comment |
up vote
0
down vote
up vote
0
down vote
I solved equation, and got an answer of $x=dfrac{23}{3}$ ,$y=dfrac{8}{3}$,$z=-5$. Is it the correct answer, by your online homework? If it is, I will share my solution with you. By the way, your answer doesn't fit the first constraint, you can check it by plugging in values.
answered Nov 21 at 11:46
Farid Hasanov
13
I solved equation, and got an answer of $x=dfrac{23}{3}$ ,$y=dfrac{8}{3}$,$z=-5$. Is it the correct answer, by your online homework? If it is, I will share my solution with you. By the way, your answer doesn't fit the first constraint, you can check it by plugging in values.
answered Nov 21 at 11:46
Farid Hasanov
13
answered Nov 21 at 11:46
Farid Hasanov
13
answered Nov 21 at 11:46
Farid Hasanov
13
answered Nov 21 at 11:46
Farid Hasanov
13
13
add a comment |
add a comment |
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2
To check whether or not you did something wrong, one would need to know what exactly you did.
– Dionel Jaime
Nov 13 at 5:11
The two constraints define a line. What are its min/max distances from the origin? In other words, check your work by solving the problem a different way.
– amd
Nov 13 at 6:26