Minimize $x^2+6y^2+4z^2$ subject to $x+2y+z-4=0$ and $2x^2+y^2=16$
Minimize $x^2+6y^2+4z^2$ subject to $x+2y+z-4=0$ and $2x^2+y^2=16$
My try:
By Lagrange Multiplier method we have
$$L(x,y,z,lambda, mu)=(x^2+6y^2+4z^2)+lambda(x+2y+z-4)+mu(2x^2+y^2-16)$$
For $$L_x=0$$ we get
$$2x+lambda+4mu x=0 tag{1}$$
For $$L_y=0$$ we get
$$12y+2lambda+2mu y=0 tag{2}$$
For $$L_z=0$$ we get
$$8z+lambda=0 tag{3}$$
From $(1)$ and $(2)$ we get
$$x=frac{4 lambda}{1-2mu}$$
$$y=frac{8 lambda}{6-mu}$$
Substituting $x$ , $y$ and $z$ above in constrainst we get
$$2 frac{lambda^2}{(1-2mu)^2}+4 frac{lambda^2}{(6-mu)^2}=1 tag{4}$$
$$frac{4 lambda}{1-2mu}+frac{16 lambda}{6-mu}+frac{lambda}{8}=4 tag{5}$$
But its tedious to solve above equations for $lambda$ and $mu$
Any other approach?
optimization systems-of-equations lagrange-multiplier maxima-minima
asked Nov 22 at 13:37
Ekaveera Kumar Sharma
5,53511328
add a comment |
Minimize $x^2+6y^2+4z^2$ subject to $x+2y+z-4=0$ and $2x^2+y^2=16$
My try:
By Lagrange Multiplier method we have
$$L(x,y,z,lambda, mu)=(x^2+6y^2+4z^2)+lambda(x+2y+z-4)+mu(2x^2+y^2-16)$$
For $$L_x=0$$ we get
$$2x+lambda+4mu x=0 tag{1}$$
For $$L_y=0$$ we get
$$12y+2lambda+2mu y=0 tag{2}$$
For $$L_z=0$$ we get
$$8z+lambda=0 tag{3}$$
From $(1)$ and $(2)$ we get
$$x=frac{4 lambda}{1-2mu}$$
$$y=frac{8 lambda}{6-mu}$$
Substituting $x$ , $y$ and $z$ above in constrainst we get
$$2 frac{lambda^2}{(1-2mu)^2}+4 frac{lambda^2}{(6-mu)^2}=1 tag{4}$$
$$frac{4 lambda}{1-2mu}+frac{16 lambda}{6-mu}+frac{lambda}{8}=4 tag{5}$$
But its tedious to solve above equations for $lambda$ and $mu$
Any other approach?
optimization systems-of-equations lagrange-multiplier maxima-minima
asked Nov 22 at 13:37
Ekaveera Kumar Sharma
5,53511328
WLOG $sqrt2x=4cos t,y=4sin t, z=?$
– lab bhattacharjee
Nov 22 at 13:43
Good idea sir. we have $z=4-2sqrt{2} cos t-8 sin t$. To minimize $f(t)=8cos^2 t+96 sin^2 t+4left(4-2sqrt{2} cos t-8 sin t right)^2$. But this also seems lengthy
– Ekaveera Kumar Sharma
Nov 22 at 13:58
Try to solve equations $(4)$ and $(5)$ by setting $text{sin}(t) = frac{sqrt{2}lambda}{1-2mu}$ and $text{cos}(t) = frac{2lambda}{6-mu}$. Hence, use trigonometric identities to reduce the problem to a quadratic equation on $text{tan}(t)$.
– Alex Silva
Nov 22 at 15:29
add a comment |
Minimize $x^2+6y^2+4z^2$ subject to $x+2y+z-4=0$ and $2x^2+y^2=16$
My try:
By Lagrange Multiplier method we have
$$L(x,y,z,lambda, mu)=(x^2+6y^2+4z^2)+lambda(x+2y+z-4)+mu(2x^2+y^2-16)$$
For $$L_x=0$$ we get
$$2x+lambda+4mu x=0 tag{1}$$
For $$L_y=0$$ we get
$$12y+2lambda+2mu y=0 tag{2}$$
For $$L_z=0$$ we get
$$8z+lambda=0 tag{3}$$
From $(1)$ and $(2)$ we get
$$x=frac{4 lambda}{1-2mu}$$
$$y=frac{8 lambda}{6-mu}$$
Substituting $x$ , $y$ and $z$ above in constrainst we get
$$2 frac{lambda^2}{(1-2mu)^2}+4 frac{lambda^2}{(6-mu)^2}=1 tag{4}$$
$$frac{4 lambda}{1-2mu}+frac{16 lambda}{6-mu}+frac{lambda}{8}=4 tag{5}$$
But its tedious to solve above equations for $lambda$ and $mu$
Any other approach?
optimization systems-of-equations lagrange-multiplier maxima-minima
asked Nov 22 at 13:37
Ekaveera Kumar Sharma
5,53511328
Minimize $x^2+6y^2+4z^2$ subject to $x+2y+z-4=0$ and $2x^2+y^2=16$
My try:
By Lagrange Multiplier method we have
$$L(x,y,z,lambda, mu)=(x^2+6y^2+4z^2)+lambda(x+2y+z-4)+mu(2x^2+y^2-16)$$
For $$L_x=0$$ we get
$$2x+lambda+4mu x=0 tag{1}$$
For $$L_y=0$$ we get
$$12y+2lambda+2mu y=0 tag{2}$$
For $$L_z=0$$ we get
$$8z+lambda=0 tag{3}$$
From $(1)$ and $(2)$ we get
$$x=frac{4 lambda}{1-2mu}$$
$$y=frac{8 lambda}{6-mu}$$
Substituting $x$ , $y$ and $z$ above in constrainst we get
$$2 frac{lambda^2}{(1-2mu)^2}+4 frac{lambda^2}{(6-mu)^2}=1 tag{4}$$
$$frac{4 lambda}{1-2mu}+frac{16 lambda}{6-mu}+frac{lambda}{8}=4 tag{5}$$
But its tedious to solve above equations for $lambda$ and $mu$
Any other approach?
optimization systems-of-equations lagrange-multiplier maxima-minima
optimization systems-of-equations lagrange-multiplier maxima-minima
asked Nov 22 at 13:37
Ekaveera Kumar Sharma
5,53511328
asked Nov 22 at 13:37
Ekaveera Kumar Sharma
5,53511328
asked Nov 22 at 13:37
Ekaveera Kumar Sharma
5,53511328
asked Nov 22 at 13:37
Ekaveera Kumar Sharma
5,53511328
asked Nov 22 at 13:37
Ekaveera Kumar Sharma
5,53511328
5,53511328
WLOG $sqrt2x=4cos t,y=4sin t, z=?$
– lab bhattacharjee
Nov 22 at 13:43
Good idea sir. we have $z=4-2sqrt{2} cos t-8 sin t$. To minimize $f(t)=8cos^2 t+96 sin^2 t+4left(4-2sqrt{2} cos t-8 sin t right)^2$. But this also seems lengthy
– Ekaveera Kumar Sharma
Nov 22 at 13:58
Try to solve equations $(4)$ and $(5)$ by setting $text{sin}(t) = frac{sqrt{2}lambda}{1-2mu}$ and $text{cos}(t) = frac{2lambda}{6-mu}$. Hence, use trigonometric identities to reduce the problem to a quadratic equation on $text{tan}(t)$.
– Alex Silva
Nov 22 at 15:29
add a comment |
WLOG $sqrt2x=4cos t,y=4sin t, z=?$
– lab bhattacharjee
Nov 22 at 13:43
Good idea sir. we have $z=4-2sqrt{2} cos t-8 sin t$. To minimize $f(t)=8cos^2 t+96 sin^2 t+4left(4-2sqrt{2} cos t-8 sin t right)^2$. But this also seems lengthy
– Ekaveera Kumar Sharma
Nov 22 at 13:58
Try to solve equations $(4)$ and $(5)$ by setting $text{sin}(t) = frac{sqrt{2}lambda}{1-2mu}$ and $text{cos}(t) = frac{2lambda}{6-mu}$. Hence, use trigonometric identities to reduce the problem to a quadratic equation on $text{tan}(t)$.
– Alex Silva
Nov 22 at 15:29
WLOG $sqrt2x=4cos t,y=4sin t, z=?$
– lab bhattacharjee
Nov 22 at 13:43
WLOG $sqrt2x=4cos t,y=4sin t, z=?$
– lab bhattacharjee
Nov 22 at 13:43
Good idea sir. we have $z=4-2sqrt{2} cos t-8 sin t$. To minimize $f(t)=8cos^2 t+96 sin^2 t+4left(4-2sqrt{2} cos t-8 sin t right)^2$. But this also seems lengthy
– Ekaveera Kumar Sharma
Nov 22 at 13:58
Good idea sir. we have $z=4-2sqrt{2} cos t-8 sin t$. To minimize $f(t)=8cos^2 t+96 sin^2 t+4left(4-2sqrt{2} cos t-8 sin t right)^2$. But this also seems lengthy
– Ekaveera Kumar Sharma
Nov 22 at 13:58
Try to solve equations $(4)$ and $(5)$ by setting $text{sin}(t) = frac{sqrt{2}lambda}{1-2mu}$ and $text{cos}(t) = frac{2lambda}{6-mu}$. Hence, use trigonometric identities to reduce the problem to a quadratic equation on $text{tan}(t)$.
– Alex Silva
Nov 22 at 15:29
Try to solve equations $(4)$ and $(5)$ by setting $text{sin}(t) = frac{sqrt{2}lambda}{1-2mu}$ and $text{cos}(t) = frac{2lambda}{6-mu}$. Hence, use trigonometric identities to reduce the problem to a quadratic equation on $text{tan}(t)$.
– Alex Silva
Nov 22 at 15:29
add a comment |
3 Answers
3
active
oldest
votes
Hint:
Let $u=x^2+6y^2+4(4-2y-x)^2$
$iff5x^2-16(2-y)x+22y^2-16y+16-u=0$
As $x$ is real, the discriminant will be $ge0$
$$implies256(y-2)^2ge20(22y^2-16y+16-u)$$
$$implies64(y-2)^2ge5(22y^2-16y+16-u)$$
$$implies5uge46y^2-176y+176=46left(y-dfrac{44}{23}right)^2+176-46left(dfrac{44}{23}right)^2$$
Now $y^2=16-2x^2le16implies-4le yle4iff-4-dfrac{44}{23}le y-dfrac{44}{23}le4-dfrac{44}{23}$
answered Nov 23 at 6:26
lab bhattacharjee
222k15155273
Very nice approach
– Ekaveera Kumar Sharma
Nov 23 at 7:10
add a comment |
Since you are assuming that $2x^2+y^2=16$, your problem is equivalent to the problem of minimizing $frac{11}2y^2+4z^2$. But then your first equation becomes $lambda+4mu x=0$, which is much simpler.
answered Nov 22 at 13:59
José Carlos Santos
148k22117218
Good ,so with your idea i got $x=frac{-lambda}{4 mu}$, $y=frac{-2lambda}{2mu+11}$, $z=frac{-lambda}{8}$. But i don't think its simple enough again?
– Ekaveera Kumar Sharma
Nov 22 at 14:29
add a comment |
With computer I get numeric solution
$$f_{min}=9.445967377367024$$
if
$$x=2.811441051354938\
y=0.4377195786259503\
z=0.3131197913931612$$
Exact values of $x,y,z$ is solutions of equations:
$$2033x^4-800x^3-13960x^2-1792x+6144=0,\
2033y^4-6016y^3-27920y^2+88064y-32768=0,\
2033z^4-19696z^3-84456z^2+176704z-46464=0.$$
We can solve these equations exact and get exact solutions.
However, there are too big expressions.
answered Nov 23 at 11:23
Aleksas Domarkas
8276
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3009146%2fminimize-x26y24z2-subject-to-x2yz-4-0-and-2x2y2-16%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
Hint:
Let $u=x^2+6y^2+4(4-2y-x)^2$
$iff5x^2-16(2-y)x+22y^2-16y+16-u=0$
As $x$ is real, the discriminant will be $ge0$
$$implies256(y-2)^2ge20(22y^2-16y+16-u)$$
$$implies64(y-2)^2ge5(22y^2-16y+16-u)$$
$$implies5uge46y^2-176y+176=46left(y-dfrac{44}{23}right)^2+176-46left(dfrac{44}{23}right)^2$$
Now $y^2=16-2x^2le16implies-4le yle4iff-4-dfrac{44}{23}le y-dfrac{44}{23}le4-dfrac{44}{23}$
answered Nov 23 at 6:26
lab bhattacharjee
222k15155273
Very nice approach
– Ekaveera Kumar Sharma
Nov 23 at 7:10
add a comment |
Hint:
Let $u=x^2+6y^2+4(4-2y-x)^2$
$iff5x^2-16(2-y)x+22y^2-16y+16-u=0$
As $x$ is real, the discriminant will be $ge0$
$$implies256(y-2)^2ge20(22y^2-16y+16-u)$$
$$implies64(y-2)^2ge5(22y^2-16y+16-u)$$
$$implies5uge46y^2-176y+176=46left(y-dfrac{44}{23}right)^2+176-46left(dfrac{44}{23}right)^2$$
Now $y^2=16-2x^2le16implies-4le yle4iff-4-dfrac{44}{23}le y-dfrac{44}{23}le4-dfrac{44}{23}$
answered Nov 23 at 6:26
lab bhattacharjee
222k15155273
Very nice approach
– Ekaveera Kumar Sharma
Nov 23 at 7:10
add a comment |
Hint:
Let $u=x^2+6y^2+4(4-2y-x)^2$
$iff5x^2-16(2-y)x+22y^2-16y+16-u=0$
As $x$ is real, the discriminant will be $ge0$
$$implies256(y-2)^2ge20(22y^2-16y+16-u)$$
$$implies64(y-2)^2ge5(22y^2-16y+16-u)$$
$$implies5uge46y^2-176y+176=46left(y-dfrac{44}{23}right)^2+176-46left(dfrac{44}{23}right)^2$$
Now $y^2=16-2x^2le16implies-4le yle4iff-4-dfrac{44}{23}le y-dfrac{44}{23}le4-dfrac{44}{23}$
answered Nov 23 at 6:26
lab bhattacharjee
222k15155273
Hint:
Let $u=x^2+6y^2+4(4-2y-x)^2$
$iff5x^2-16(2-y)x+22y^2-16y+16-u=0$
As $x$ is real, the discriminant will be $ge0$
$$implies256(y-2)^2ge20(22y^2-16y+16-u)$$
$$implies64(y-2)^2ge5(22y^2-16y+16-u)$$
$$implies5uge46y^2-176y+176=46left(y-dfrac{44}{23}right)^2+176-46left(dfrac{44}{23}right)^2$$
Now $y^2=16-2x^2le16implies-4le yle4iff-4-dfrac{44}{23}le y-dfrac{44}{23}le4-dfrac{44}{23}$
answered Nov 23 at 6:26
lab bhattacharjee
222k15155273
answered Nov 23 at 6:26
lab bhattacharjee
222k15155273
answered Nov 23 at 6:26
lab bhattacharjee
222k15155273
answered Nov 23 at 6:26
lab bhattacharjee
222k15155273
222k15155273
Very nice approach
– Ekaveera Kumar Sharma
Nov 23 at 7:10
add a comment |
Very nice approach
– Ekaveera Kumar Sharma
Nov 23 at 7:10
Very nice approach
– Ekaveera Kumar Sharma
Nov 23 at 7:10
Very nice approach
– Ekaveera Kumar Sharma
Nov 23 at 7:10
add a comment |
Since you are assuming that $2x^2+y^2=16$, your problem is equivalent to the problem of minimizing $frac{11}2y^2+4z^2$. But then your first equation becomes $lambda+4mu x=0$, which is much simpler.
answered Nov 22 at 13:59
José Carlos Santos
148k22117218
Good ,so with your idea i got $x=frac{-lambda}{4 mu}$, $y=frac{-2lambda}{2mu+11}$, $z=frac{-lambda}{8}$. But i don't think its simple enough again?
– Ekaveera Kumar Sharma
Nov 22 at 14:29
add a comment |
Since you are assuming that $2x^2+y^2=16$, your problem is equivalent to the problem of minimizing $frac{11}2y^2+4z^2$. But then your first equation becomes $lambda+4mu x=0$, which is much simpler.
answered Nov 22 at 13:59
José Carlos Santos
148k22117218
Good ,so with your idea i got $x=frac{-lambda}{4 mu}$, $y=frac{-2lambda}{2mu+11}$, $z=frac{-lambda}{8}$. But i don't think its simple enough again?
– Ekaveera Kumar Sharma
Nov 22 at 14:29
add a comment |
Since you are assuming that $2x^2+y^2=16$, your problem is equivalent to the problem of minimizing $frac{11}2y^2+4z^2$. But then your first equation becomes $lambda+4mu x=0$, which is much simpler.
answered Nov 22 at 13:59
José Carlos Santos
148k22117218
Since you are assuming that $2x^2+y^2=16$, your problem is equivalent to the problem of minimizing $frac{11}2y^2+4z^2$. But then your first equation becomes $lambda+4mu x=0$, which is much simpler.
answered Nov 22 at 13:59
José Carlos Santos
148k22117218
edited Nov 22 at 14:21
answered Nov 22 at 13:59
José Carlos Santos
148k22117218
answered Nov 22 at 13:59
José Carlos Santos
148k22117218
answered Nov 22 at 13:59
José Carlos Santos
148k22117218
148k22117218
Good ,so with your idea i got $x=frac{-lambda}{4 mu}$, $y=frac{-2lambda}{2mu+11}$, $z=frac{-lambda}{8}$. But i don't think its simple enough again?
– Ekaveera Kumar Sharma
Nov 22 at 14:29
add a comment |
Good ,so with your idea i got $x=frac{-lambda}{4 mu}$, $y=frac{-2lambda}{2mu+11}$, $z=frac{-lambda}{8}$. But i don't think its simple enough again?
– Ekaveera Kumar Sharma
Nov 22 at 14:29
Good ,so with your idea i got $x=frac{-lambda}{4 mu}$, $y=frac{-2lambda}{2mu+11}$, $z=frac{-lambda}{8}$. But i don't think its simple enough again?
– Ekaveera Kumar Sharma
Nov 22 at 14:29
Good ,so with your idea i got $x=frac{-lambda}{4 mu}$, $y=frac{-2lambda}{2mu+11}$, $z=frac{-lambda}{8}$. But i don't think its simple enough again?
– Ekaveera Kumar Sharma
Nov 22 at 14:29
add a comment |
With computer I get numeric solution
$$f_{min}=9.445967377367024$$
if
$$x=2.811441051354938\
y=0.4377195786259503\
z=0.3131197913931612$$
Exact values of $x,y,z$ is solutions of equations:
$$2033x^4-800x^3-13960x^2-1792x+6144=0,\
2033y^4-6016y^3-27920y^2+88064y-32768=0,\
2033z^4-19696z^3-84456z^2+176704z-46464=0.$$
We can solve these equations exact and get exact solutions.
However, there are too big expressions.
answered Nov 23 at 11:23
Aleksas Domarkas
8276
add a comment |
With computer I get numeric solution
$$f_{min}=9.445967377367024$$
if
$$x=2.811441051354938\
y=0.4377195786259503\
z=0.3131197913931612$$
Exact values of $x,y,z$ is solutions of equations:
$$2033x^4-800x^3-13960x^2-1792x+6144=0,\
2033y^4-6016y^3-27920y^2+88064y-32768=0,\
2033z^4-19696z^3-84456z^2+176704z-46464=0.$$
We can solve these equations exact and get exact solutions.
However, there are too big expressions.
answered Nov 23 at 11:23
Aleksas Domarkas
8276
add a comment |
With computer I get numeric solution
$$f_{min}=9.445967377367024$$
if
$$x=2.811441051354938\
y=0.4377195786259503\
z=0.3131197913931612$$
Exact values of $x,y,z$ is solutions of equations:
$$2033x^4-800x^3-13960x^2-1792x+6144=0,\
2033y^4-6016y^3-27920y^2+88064y-32768=0,\
2033z^4-19696z^3-84456z^2+176704z-46464=0.$$
We can solve these equations exact and get exact solutions.
However, there are too big expressions.
answered Nov 23 at 11:23
Aleksas Domarkas
8276
With computer I get numeric solution
$$f_{min}=9.445967377367024$$
if
$$x=2.811441051354938\
y=0.4377195786259503\
z=0.3131197913931612$$
Exact values of $x,y,z$ is solutions of equations:
$$2033x^4-800x^3-13960x^2-1792x+6144=0,\
2033y^4-6016y^3-27920y^2+88064y-32768=0,\
2033z^4-19696z^3-84456z^2+176704z-46464=0.$$
We can solve these equations exact and get exact solutions.
However, there are too big expressions.
answered Nov 23 at 11:23
Aleksas Domarkas
8276
answered Nov 23 at 11:23
Aleksas Domarkas
8276
answered Nov 23 at 11:23
Aleksas Domarkas
8276
answered Nov 23 at 11:23
Aleksas Domarkas
8276
8276
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3009146%2fminimize-x26y24z2-subject-to-x2yz-4-0-and-2x2y2-16%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
WLOG $sqrt2x=4cos t,y=4sin t, z=?$
– lab bhattacharjee
Nov 22 at 13:43
Good idea sir. we have $z=4-2sqrt{2} cos t-8 sin t$. To minimize $f(t)=8cos^2 t+96 sin^2 t+4left(4-2sqrt{2} cos t-8 sin t right)^2$. But this also seems lengthy
– Ekaveera Kumar Sharma
Nov 22 at 13:58
Try to solve equations $(4)$ and $(5)$ by setting $text{sin}(t) = frac{sqrt{2}lambda}{1-2mu}$ and $text{cos}(t) = frac{2lambda}{6-mu}$. Hence, use trigonometric identities to reduce the problem to a quadratic equation on $text{tan}(t)$.
– Alex Silva
Nov 22 at 15:29