Question Number 85241 by Power last updated on 20/Mar/20
Answered by jagoll last updated on 20/Mar/20
$$\mathrm{minimum}\:\frac{\mathrm{4s}^{\mathrm{2}} +\mathrm{t}^{\mathrm{2}} }{\mathrm{5st}}\:=\:\frac{\mathrm{4s}^{\mathrm{2}} }{\mathrm{5st}}\:+\:\frac{\mathrm{t}^{\mathrm{2}} }{\mathrm{5st}} \\ $$$$=\:\frac{\mathrm{4}}{\mathrm{5}}\left(\frac{\mathrm{s}}{\mathrm{t}}\right)\:+\:\frac{\mathrm{1}}{\mathrm{5}}\left(\frac{\mathrm{t}}{\mathrm{s}}\right). \\ $$$$\mathrm{let}\:\frac{\mathrm{s}}{\mathrm{t}}\:=\:\mathrm{u}\:\Rightarrow\mathrm{f}\left(\mathrm{u}\right)\:=\:\frac{\mathrm{4}}{\mathrm{5}}\mathrm{u}+\frac{\mathrm{1}}{\mathrm{5}}\mathrm{u}^{−\mathrm{1}} \\ $$$$\mathrm{f}\:'\left(\mathrm{u}\right)=\:\frac{\mathrm{4}}{\mathrm{5}}−\frac{\mathrm{1}}{\mathrm{5u}^{\mathrm{2}} }\:=\:\mathrm{0}\Rightarrow\:\mathrm{u}\:=\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\mathrm{minimum}\:\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\:=\:\frac{\mathrm{4}}{\mathrm{5}}×\frac{\mathrm{1}}{\mathrm{2}}+\:\frac{\mathrm{1}}{\mathrm{5}}×\mathrm{2} \\ $$$$=\frac{\mathrm{2}}{\mathrm{5}}+\frac{\mathrm{2}}{\mathrm{5}}=\:\frac{\mathrm{4}}{\mathrm{5}} \\ $$
Commented by mr W last updated on 20/Mar/20
$${we}\:{can}\:{get}\:{this}\:{result}\:{without}\:{calculus}. \\ $$$${we}\:{knowfor}\:{positive}\:{A}\:{and}\:{B}: \\ $$$${A}^{\mathrm{2}} +{B}^{\mathrm{2}} −\mathrm{2}{AB}=\left({A}−{B}\right)^{\mathrm{2}} \geqslant\mathrm{0} \\ $$$$\Rightarrow{A}^{\mathrm{2}} +{B}^{\mathrm{2}} \geqslant\mathrm{2}{AB} \\ $$$${now}\:{with}\:{A}=\mathrm{2}{s},\:{B}={t} \\ $$$$\frac{\mathrm{4s}^{\mathrm{2}} +\mathrm{t}^{\mathrm{2}} }{\mathrm{5st}}=\frac{\left(\mathrm{2}{s}\right)^{\mathrm{2}} +\mathrm{t}^{\mathrm{2}} }{\mathrm{5st}}\geqslant\frac{\mathrm{2}\left(\mathrm{2}{s}\right){t}}{\mathrm{5}{st}}=\frac{\mathrm{4}}{\mathrm{5}} \\ $$$${i}.{e}.\:{the}\:{minimum}\:{of}\:\frac{\mathrm{4s}^{\mathrm{2}} +\mathrm{t}^{\mathrm{2}} }{\mathrm{5st}}\:{is}\:\frac{\mathrm{4}}{\mathrm{5}}. \\ $$
Commented by jagoll last updated on 20/Mar/20
$$\mathrm{good}\:\mathrm{improvment} \\ $$
Commented by Power last updated on 20/Mar/20
$$\mathrm{thanks} \\ $$