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If-a-b-c-lt-0-and-abc-a-b-c-64-Then-find-min-of-P-2a-b-c-




Question Number 180873 by Shrinava last updated on 18/Nov/22
If   a,b,c<0   and   abc(a+b+c)=64  Then find min of   P=2a+b+c
Ifa,b,c<0andabc(a+b+c)=64ThenfindminofP=2a+b+c
Commented by mr W last updated on 18/Nov/22
please check the question. since  a,b,c<0, P has maximum, not min.  or you mean a,b,c>0?
pleasecheckthequestion.sincea,b,c<0,Phasmaximum,notmin.oryoumeana,b,c>0?
Commented by Shrinava last updated on 18/Nov/22
sorry dear professor >0
sorrydearprofessor>0
Answered by mr W last updated on 18/Nov/22
due to symmetry, P has extremum  when b=c=k, say  P=2a+b+c=2(a+k)  ak^2 (a+2k)=64  k^2 a^2 +2k^3 a−64=0  ⇒a=(√(k^2 +((64)/k^2 )))−k  ⇒P=2(√(k^2 +((64)/k^2 )))≥2(√(2×8))=8=P_(min)
duetosymmetry,Phasextremumwhenb=c=k,sayP=2a+b+c=2(a+k)ak2(a+2k)=64k2a2+2k3a64=0a=k2+64k2kP=2k2+64k222×8=8=Pmin
Commented by Shrinava last updated on 18/Nov/22
Thank you so much dear professor
Thankyousomuchdearprofessor

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