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Question Number 188535 by Rupesh123 last updated on 03/Mar/23

Answered by mr W last updated on 03/Mar/23

due to symmetry  at extremum  a=b=c=x>0  S=((3(x+1)^3 )/x)=((3(x+(1/2)+(1/2)))/x)                          ≥((3(3((x×(1/2)×(1/2)))^(1/3) )^3 )/x)                          =3×27×(1/4)=((81)/4)  ⇒minimum=((81)/4)

$${due}\:{to}\:{symmetry} \\ $$$${at}\:{extremum} \\ $$$${a}={b}={c}={x}>\mathrm{0} \\ $$$${S}=\frac{\mathrm{3}\left({x}+\mathrm{1}\right)^{\mathrm{3}} }{{x}}=\frac{\mathrm{3}\left({x}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\right)}{{x}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\geqslant\frac{\mathrm{3}\left(\mathrm{3}\sqrt[{\mathrm{3}}]{{x}×\frac{\mathrm{1}}{\mathrm{2}}×\frac{\mathrm{1}}{\mathrm{2}}}\right)^{\mathrm{3}} }{{x}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{3}×\mathrm{27}×\frac{\mathrm{1}}{\mathrm{4}}=\frac{\mathrm{81}}{\mathrm{4}} \\ $$$$\Rightarrow{minimum}=\frac{\mathrm{81}}{\mathrm{4}} \\ $$

Commented by Rupesh123 last updated on 03/Mar/23

Excellent

Commented by cortano12 last updated on 04/Mar/23

 ((3(x+1)^3 )/x)=((3(x+(1/2)+(1/2))^3 )/x) ?

$$\:\frac{\mathrm{3}\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{3}} }{\mathrm{x}}=\frac{\mathrm{3}\left(\mathrm{x}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{3}} }{\mathrm{x}}\:? \\ $$

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