Question Number 44898 by ajfour last updated on 06/Oct/18
$${If}\:\:\:\:{x}^{\mathrm{4}} +{px}^{\mathrm{3}} +{qx}^{\mathrm{2}} +{rx}+\mathrm{5}\:=\:\mathrm{0} \\ $$$${has}\:{four}\:{real}\:{roots},\:{then}\:{find} \\ $$$$\:{the}\:{minimum}\:{value}\:{of}\:\boldsymbol{{pr}}. \\ $$
Commented by MJS last updated on 06/Oct/18
$$\mathrm{let}\:\mathrm{me}\:\mathrm{try}... \\ $$
Answered by ajfour last updated on 06/Oct/18
$${x}_{\mathrm{1}} {x}_{\mathrm{2}} {x}_{\mathrm{3}} {x}_{\mathrm{4}} =\:\mathrm{5} \\ $$$$\Sigma{x}_{\mathrm{1}} {x}_{\mathrm{2}} {x}_{\mathrm{3}} \:=\:−{r} \\ $$$${x}_{\mathrm{1}} +{x}_{\mathrm{2}} +{x}_{\mathrm{3}} +{x}_{\mathrm{4}} \:=\:−{p} \\ $$$${pr}\:=\:\mathrm{5}\left(\frac{\mathrm{1}}{{x}_{\mathrm{4}} }+\frac{\mathrm{1}}{{x}_{\mathrm{2}} }+\frac{\mathrm{1}}{{x}_{\mathrm{3}} }+\frac{\mathrm{1}}{{x}_{\mathrm{4}} }\right)\Sigma\:{x}_{\mathrm{1}} \: \\ $$$$\:\:\:\:>\:\mathrm{5}×\frac{\mathrm{4}}{{x}_{{c}} }×\mathrm{4}{x}_{{c}} \:\:\:\left(=\:\mathrm{80}\right)\:. \\ $$$${Hence}\:{min}\left(\boldsymbol{{pr}}\right)\:=\:\mathrm{80}\:. \\ $$$$ \\ $$