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Question Number 134817 by bramlexs22 last updated on 07/Mar/21
What is the maximum value of abc if a+b+c = 5 and ab+bc+ca = 7?
What is the maximum value of abc if a+b+c = 5 and ab+bc+ca = 7?
Commented by bramlexs22 last updated on 09/Mar/21
thanks you all sir
thanksyouallsir
Commented by john_santu last updated on 08/Mar/21
let a+b = x then c = 5−x (i) xc = (a+b)c = ac+bc = 7−ab then ab = 7−xc so abc = (7−xc)c abc = 7c−xc^2 , injecting c=5−x we get abc = f(x)= 7(5−x)−x(5−x)^2 f(x)= 35−7x−x(x^2 −10x+25) f(x)=−x^3 +10x^2 −32x+35 (df/dx) = −3x^2 +20x−32 = 0 we get { ((x=(8/3))),((x=4)) :} (d^2 f/dx^2 ) = −6x+20∣_(x= 4) <0 and (d^2 f/dx^2 ) = −6x+20∣_(x = (8/3)) = −((48)/3)+20 >0 so f(x)_(max) for x = 4 and f(x)_(min ) for x=(8/3) ∴ max value is f(4)=−4^3 +10.4^2 −32.4+35 f(4)= −64+160−128+35 = 3 when a+b=4 and c = 1 . from eq(2) ab+ac+bc = 7⇒ab+a+b = 7 ; ab = 3 ∧b = 4−a ; we get a(4−a)=3 a^2 −4a+3 = 0 → { ((a=1→b=3 or)),((a=3→b=1 )) :} (a,b,c) = (1,3,1) or (3,1,1) clearly max is f(4) = 3 min value is f((8/3))=−((8/3))^3 +10((8/3))^2 −32((8/3))+35
leta+b=xthenc=5−x(i)xc=(a+b)c=ac+bc=7−abthenab=7−xcsoabc=(7−xc)cabc=7c−xc2,injectingc=5−xwegetabc=f(x)=7(5−x)−x(5−x)2f(x)=35−7x−x(x2−10x+25)f(x)=−x3+10x2−32x+35dfdx=−3x2+20x−32=0weget{x=83x=4d2fdx2=−6x+20∣x=4<0andd2fdx2=−6x+20∣x=83=−483+20>0sof(x)maxforx=4andf(x)minforx=83∴maxvalueisf(4)=−43+10.42−32.4+35f(4)=−64+160−128+35=3whena+b=4andc=1.fromeq(2)ab+ac+bc=7⇒ab+a+b=7;ab=3∧b=4−a;wegeta(4−a)=3a2−4a+3=0→{a=1→b=3ora=3→b=1(a,b,c)=(1,3,1)or(3,1,1)clearlymaxisf(4)=3minvalueisf(83)=−(83)3+10(83)2−32(83)+35
Commented by mr W last updated on 08/Mar/21
very nice! i used also similar method, but maybe made a calculation mistake somewhere.
verynice!iusedalsosimilarmethod,butmaybemadeacalculationmistakesomewhere.
Answered by liberty last updated on 07/Mar/21
(1)a=5−(b+c) (2) (5−(b+c))b+bc+c(5−(b+c))=7 ⇒ 5b−b^2 −bc+bc+5c−bc−c^2 =7 ⇒5b−b^2 −bc+5c+c^2 = 7 (3)abc = (5−(b+c))bc =5bc−b^2 c−bc^2 let f(b,c)=5bc−b^2 c−bc^2 +λ(5b−b^2 −bc+5c+c^2 −7) (i) f_b = 5c−2bc−c^2 +λ(5−2b−c)=0 (ii)f_c = 5b−b^2 −2bc+λ(−b+5+2c)= 0 (iii)f_(λ ) = 5b−b^2 −bc+5c+c^2 = 7 from eq (i) &(ii) we get { ((λ=((c^2 +2bc−5c)/(5−2b−c)))),((λ = ((b^2 +2bc−5b)/(5−b+2c)))) :} and b^2 −5b = c^2 +5c−bc−7 ⇒ λ = λ ; ((c^2 +2bc−5c)/(5−2b−c)) = ((c^2 +5c−bc−7+2bc)/(5−b+2c)) next...
(1)a=5−(b+c)(2)(5−(b+c))b+bc+c(5−(b+c))=7⇒5b−b2−bc+bc+5c−bc−c2=7⇒5b−b2−bc+5c+c2=7(3)abc=(5−(b+c))bc=5bc−b2c−bc2letf(b,c)=5bc−b2c−bc2+λ(5b−b2−bc+5c+c2−7)(i)fb=5c−2bc−c2+λ(5−2b−c)=0(ii)fc=5b−b2−2bc+λ(−b+5+2c)=0(iii)fλ=5b−b2−bc+5c+c2=7fromeq(i)&(ii)weget{λ=c2+2bc−5c5−2b−cλ=b2+2bc−5b5−b+2candb2−5b=c2+5c−bc−7⇒λ=λ;c2+2bc−5c5−2b−c=c2+5c−bc−7+2bc5−b+2cnext…
Answered by mr W last updated on 08/Mar/21
c=5−(a+b) ab+(a+b)c=7 c=((7−ab)/(a+b))=5−(a+b) ⇒ab=7−(a+b)(5−(a+b)) P=abc=ab(5−(a+b))=[7−(a+b)(5−(a+b))](5−(a+b)) let t=a+b ⇒P=[7−t(5−t)](5−t)=−t^3 +10t^2 −32t+35 (dP/dt)=−3t^2 +20t−32=0 (3t−8)(t−4)=0 ⇒t=(8/3), 4 (d^2 P/dt^2 )=−6t+20 at t=(8/3): (d^2 P/dt^2 )=−6×(8/3)+20=4>0 ⇒min. at t=4: (d^2 P/dt^2 )=−6×4+20=−4<0 ⇒max. ⇒P_(min) =[7−(8/3)(5−(8/3))](5−(8/3))=((49)/(27)) ⇒P_(max) =[7−4(5−4)](5−4)=3
c=5−(a+b)ab+(a+b)c=7c=7−aba+b=5−(a+b)⇒ab=7−(a+b)(5−(a+b))P=abc=ab(5−(a+b))=[7−(a+b)(5−(a+b))](5−(a+b))lett=a+b⇒P=[7−t(5−t)](5−t)=−t3+10t2−32t+35dPdt=−3t2+20t−32=0(3t−8)(t−4)=0⇒t=83,4d2Pdt2=−6t+20att=83:d2Pdt2=−6×83+20=4>0⇒min.att=4:d2Pdt2=−6×4+20=−4<0⇒max.⇒Pmin=[7−83(5−83)](5−83)=4927⇒Pmax=[7−4(5−4)](5−4)=3
Answered by ajfour last updated on 08/Mar/21
x^3 −5x^2 +7x−m=0 where a+b+c=5 ab+bc+ca=7 abc=m m=x^3 −5x^2 +7x (dm/dx)=3x^2 −10x+7 x=((10±(√(100−84)))/6) x=((10±4)/6) ⇒ x=1, (7/3) for x=1, m=3 for x=(7/3) ,m=((49)/(27)) hence (abc)_(max) = m_(max) =3
x3−5x2+7x−m=0wherea+b+c=5ab+bc+ca=7abc=mm=x3−5x2+7xdmdx=3x2−10x+7x=10±100−846x=10±46⇒x=1,73forx=1,m=3forx=73,m=4927hence(abc)max=mmax=3
Commented by mr W last updated on 09/Mar/21
nice idea!
niceidea!
Commented by ajfour last updated on 09/Mar/21
thanks sir, m little shaken by circumstances, n anyway taken to watching Utube movies n music, cant focus here like i used to, lets see how long; I m so very unpredictable even to myself...
thankssir,mlittleshakenbycircumstances,nanywaytakentowatchingUtubemoviesnmusic,cantfocusherelikeiusedto,letsseehowlong;Imsoveryunpredictableeventomyself…
Commented by mr W last updated on 09/Mar/21
take best care of yourself sir!
takebestcareofyourselfsir!

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