Minimize $x^2+6y^2+4z^2$ subject to $x+2y+z-4=0$ and $2x^2+y^2=16$











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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







share|cite|improve this question






















  • 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

















up vote
2
down vote

favorite
2












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







share|cite|improve this question






















  • 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















up vote
2
down vote

favorite
2









up vote
2
down vote

favorite
2






2





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







share|cite|improve this question













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






share|cite|improve this question













share|cite|improve this question











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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




















  • 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












3 Answers
3






active

oldest

votes

















up vote
1
down vote



accepted










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 $y^2=16-2x^2le16implies-4le yle4iff-4-dfrac{44}{23}le y-dfrac{44}{23}le4-dfrac{44}{23}$






share|cite|improve this answer





















  • Very nice approach
    – Ekaveera Kumar Sharma
    Nov 23 at 7:10


















up vote
1
down vote













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.






share|cite|improve this answer























  • 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


















up vote
0
down vote













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.






share|cite|improve this answer





















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    3 Answers
    3






    active

    oldest

    votes








    3 Answers
    3






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    1
    down vote



    accepted










    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 $y^2=16-2x^2le16implies-4le yle4iff-4-dfrac{44}{23}le y-dfrac{44}{23}le4-dfrac{44}{23}$






    share|cite|improve this answer





















    • Very nice approach
      – Ekaveera Kumar Sharma
      Nov 23 at 7:10















    up vote
    1
    down vote



    accepted










    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 $y^2=16-2x^2le16implies-4le yle4iff-4-dfrac{44}{23}le y-dfrac{44}{23}le4-dfrac{44}{23}$






    share|cite|improve this answer





















    • Very nice approach
      – Ekaveera Kumar Sharma
      Nov 23 at 7:10













    up vote
    1
    down vote



    accepted







    up vote
    1
    down vote



    accepted






    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 $y^2=16-2x^2le16implies-4le yle4iff-4-dfrac{44}{23}le y-dfrac{44}{23}le4-dfrac{44}{23}$






    share|cite|improve this answer












    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 $y^2=16-2x^2le16implies-4le yle4iff-4-dfrac{44}{23}le y-dfrac{44}{23}le4-dfrac{44}{23}$







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Nov 23 at 6:26









    lab bhattacharjee

    222k15155273




    222k15155273












    • 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




    Very nice approach
    – Ekaveera Kumar Sharma
    Nov 23 at 7:10










    up vote
    1
    down vote













    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.






    share|cite|improve this answer























    • 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















    up vote
    1
    down vote













    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.






    share|cite|improve this answer























    • 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













    up vote
    1
    down vote










    up vote
    1
    down vote









    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.






    share|cite|improve this answer














    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.







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited Nov 22 at 14:21

























    answered Nov 22 at 13:59









    José Carlos Santos

    146k22117217




    146k22117217












    • 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




    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










    up vote
    0
    down vote













    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.






    share|cite|improve this answer

























      up vote
      0
      down vote













      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.






      share|cite|improve this answer























        up vote
        0
        down vote










        up vote
        0
        down vote









        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.






        share|cite|improve this answer












        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.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 23 at 11:23









        Aleksas Domarkas

        8186




        8186






























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