Question Number 111699 by bemath last updated on 04/Sep/20
$${find}\:{minimum}\:{value}\:{of}\:{function} \\ $$$${f}\left({x}\right)\:=\:\frac{\mathrm{27}}{\mathrm{2}{x}^{\mathrm{2}} }\:+\:\frac{\mathrm{96}}{\mathrm{27}}\:{x}^{\mathrm{2}} \\ $$
Answered by john santu last updated on 04/Sep/20
Answered by ajfour last updated on 04/Sep/20
$${f}\left({x}\right)={u}+{v} \\ $$$${uv}={c}=\mathrm{48} \\ $$$${f}\left({x}\right)\mid_{{min}} ={mn}\left(\sqrt{\left({u}−{v}\right)^{\mathrm{2}} +\mathrm{4}{uv}}\:\right)\:=\:\mathrm{2}\sqrt{{uv}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{2}\sqrt{\mathrm{48}}\:=\mathrm{8}\sqrt{\mathrm{3}}\:. \\ $$
Commented by bemath last updated on 04/Sep/20
$${how}\:{you}\:{got}\:{uv}\:=\:{c}\:=\:\mathrm{48} \\ $$
Commented by ajfour last updated on 04/Sep/20
$$\:\:\:{f}\left({x}\right)={u}+{v} \\ $$$${uv}=\frac{\mathrm{27}}{\mathrm{2}{x}^{\mathrm{2}} }×\frac{\mathrm{96}{x}^{\mathrm{2}} }{\mathrm{27}}\:=\:\mathrm{48} \\ $$
Answered by 1549442205PVT last updated on 04/Sep/20
$$\mathrm{Applying}\:\mathrm{Cauchy}'\mathrm{s}\:\mathrm{inequality}\:\mathrm{for} \\ $$$$\mathrm{positive}\:\mathrm{numbers}\:\mathrm{we}\:\mathrm{have} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{27}}{\mathrm{2}{x}^{\mathrm{2}} }\:+\:\frac{\mathrm{96}}{\mathrm{27}}\:{x}^{\mathrm{2}} \geqslant\mathrm{2}\sqrt{\frac{\mathrm{27}}{\mathrm{2x}^{\mathrm{2}} }.\frac{\mathrm{96x}^{\mathrm{2}} }{\mathrm{27}}} \\ $$$$=\mathrm{2}\sqrt{\mathrm{48}}=\mathrm{8}\sqrt{\mathrm{3}} \\ $$$$\mathrm{The}\:\mathrm{equality}\:\mathrm{ocurrs}\:\mathrm{if}\:\mathrm{and}\:\mathrm{only}\:\mathrm{if}\: \\ $$$$\frac{\mathrm{27}}{\mathrm{2x}^{\mathrm{2}} }=\frac{\mathrm{96x}^{\mathrm{2}} }{\mathrm{27}}\Leftrightarrow\mathrm{x}^{\mathrm{4}} =\frac{\mathrm{27}^{\mathrm{2}} }{\mathrm{192}}\Leftrightarrow\mathrm{x}^{\mathrm{2}} =\frac{\mathrm{27}}{\mathrm{8}\sqrt{\mathrm{3}}} \\ $$$$\Leftrightarrow\mid\mathrm{x}\mid=\frac{\mathrm{3}\sqrt{\mathrm{3}}}{\:\sqrt{\mathrm{8}\sqrt{\mathrm{3}}}}\Leftrightarrow\mathrm{x}=\pm\frac{\mathrm{3}\sqrt{\mathrm{3}}}{\:\sqrt{\mathrm{8}\sqrt{\mathrm{3}}}} \\ $$$$\mathrm{Thus},\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{has}\:\mathrm{smallest}\:\mathrm{value}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\mathrm{8}\sqrt{\mathrm{3}\:}\:\mathrm{when}\:\mathrm{x}=\pm\frac{\mathrm{3}\sqrt{\mathrm{3}}}{\:\sqrt{\mathrm{8}\sqrt{\mathrm{3}}}} \\ $$