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If-a-b-c-5-1-a-1-b-1-c-1-5-what-is-the-greatest-value-of-a-3-b-3-c-3-

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Question Number 140275 by liberty last updated on 06/May/21
If { ((a+b+c = 5)),(((1/a)+(1/b)+(1/c)=(1/5))) :} what is the greatest value of a^3 +b^3 +c^3 ?
$$\mathrm{If}\:\begin{cases}{\mathrm{a}+\mathrm{b}+\mathrm{c}\:=\:\mathrm{5}}\\{\frac{\mathrm{1}}{\mathrm{a}}+\frac{\mathrm{1}}{\mathrm{b}}+\frac{\mathrm{1}}{\mathrm{c}}=\frac{\mathrm{1}}{\mathrm{5}}}\end{cases} \\ $$$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} \:? \\ $$
Answered by Ar Brandon last updated on 06/May/21
(1/a)+(1/b)+(1/c)=(1/5)⇒((ab+bc+ca)/(abc))=(1/5) a^3 +b^3 +c^3 −3abc=(a+b+c)((a+b+c)^2 −3ab−3bc−3ca) a^3 +b^3 +c^3 =5(5^2 −3(((abc)/5)))+3abc =5^3 −3abc+3abc=125
$$\frac{\mathrm{1}}{\mathrm{a}}+\frac{\mathrm{1}}{\mathrm{b}}+\frac{\mathrm{1}}{\mathrm{c}}=\frac{\mathrm{1}}{\mathrm{5}}\Rightarrow\frac{\mathrm{ab}+\mathrm{bc}+\mathrm{ca}}{\mathrm{abc}}=\frac{\mathrm{1}}{\mathrm{5}} \\ $$$$\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} −\mathrm{3abc}=\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right)\left(\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right)^{\mathrm{2}} −\mathrm{3ab}−\mathrm{3bc}−\mathrm{3ca}\right) \\ $$$$\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} =\mathrm{5}\left(\mathrm{5}^{\mathrm{2}} −\mathrm{3}\left(\frac{\mathrm{abc}}{\mathrm{5}}\right)\right)+\mathrm{3abc} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{5}^{\mathrm{3}} −\mathrm{3abc}+\mathrm{3abc}=\mathrm{125} \\ $$

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