Question Number 44384 by Joel578 last updated on 28/Sep/18
$$\mathrm{Let}\:{a}\:\mathrm{and}\:{b}\:\mathrm{are}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{such}\:\mathrm{that} \\ $$$${a}\:>\:{b}\:>\:\mathrm{0} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of} \\ $$$$\sqrt{\mathrm{2}}{a}^{\mathrm{3}} \:+\:\frac{\mathrm{3}}{{ab}\:−\:{b}^{\mathrm{2}} } \\ $$
Commented by behi83417@gmail.com last updated on 28/Sep/18
$${f}\left({a},{b}\right)=\sqrt{\mathrm{2}}{a}^{\mathrm{3}} +\frac{\mathrm{3}}{{ab}−{b}^{\mathrm{2}} } \\ $$$$\frac{\partial{f}}{\partial{a}}=\mathrm{3}\sqrt{\mathrm{2}}{a}^{\mathrm{2}} −\frac{\mathrm{3}{b}}{\left({ab}−{b}^{\mathrm{2}} \right)^{\mathrm{2}} }=\mathrm{0}\:\:\:\left({i}\right) \\ $$$$\frac{\partial{f}}{\partial{b}}=−\frac{\mathrm{3}\left({a}−\mathrm{2}{b}\right)}{\left({ab}−{b}^{\mathrm{2}} \right)^{\mathrm{2}} }=\mathrm{0}\:\:\:\left({ii}\right) \\ $$$$\left({ii}\right)\Rightarrow{a}−\mathrm{2}{b}=\mathrm{0}\Rightarrow{a}=\mathrm{2}{b} \\ $$$$\left({i}\right)\Rightarrow\mathrm{12}\sqrt{\mathrm{2}}{b}^{\mathrm{2}} −\frac{\mathrm{3}{b}}{\left(\mathrm{2}{b}^{\mathrm{2}} −{b}^{\mathrm{2}} \right)^{\mathrm{2}} }=\mathrm{0}\Rightarrow \\ $$$$\mathrm{12}\sqrt{\mathrm{2}}{b}^{\mathrm{2}} −\frac{\mathrm{3}}{{b}^{\mathrm{3}} }=\mathrm{0}\Rightarrow{b}^{\mathrm{5}} =\frac{\mathrm{1}}{\mathrm{4}\sqrt{\mathrm{2}}}=\frac{\mathrm{1}}{\mathrm{2}^{\frac{\mathrm{5}}{\mathrm{2}}} }=\frac{\mathrm{1}}{\left(\sqrt{\mathrm{2}}\right)^{\mathrm{5}} } \\ $$$$\Rightarrow{b}=\frac{\sqrt{\mathrm{2}}}{\mathrm{2}},{a}=\sqrt{\mathrm{2}} \\ $$$$\Rightarrow{f}_{{min}} =\sqrt{\mathrm{2}}\left(\sqrt{\mathrm{2}}\right)^{\mathrm{3}} +\frac{\mathrm{3}}{\:\sqrt{\mathrm{2}}.\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{2}}}= \\ $$$$=\mathrm{4}+\frac{\mathrm{3}}{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}}=\mathrm{10}\:\:\:. \\ $$
Commented by ajfour last updated on 28/Sep/18
$${Very}\:{grateful}\:{Sir}! \\ $$
Commented by rahul 19 last updated on 28/Sep/18
$${Thanks}\:{to}\:{Mrw}_{\mathrm{3}} \:{sir}\:{for}\:{sharing}\:{this} \\ $$$${method}\:{with}\:{us}!! \\ $$
Commented by Joel578 last updated on 28/Sep/18
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much} \\ $$
Commented by tanmay.chaudhury50@gmail.com last updated on 28/Sep/18
Commented by tanmay.chaudhury50@gmail.com last updated on 28/Sep/18
Answered by ajfour last updated on 28/Sep/18
$${let}\:\:{x}=\:\sqrt{\mathrm{2}}{a}^{\mathrm{3}} +\frac{\mathrm{3}}{{b}\left({a}−{b}\right)} \\ $$$$\Rightarrow\:\:{b}\left({a}−{b}\right)=\frac{\mathrm{3}}{\left({x}−\sqrt{\mathrm{2}}{a}^{\mathrm{3}} \right)} \\ $$$${if}\:{x}\:{is}\:{minimum},\:{b}\left({a}−{b}\right)\:{is}\:{maximum} \\ $$$$\Rightarrow\:{b}\:=\:\frac{{a}}{\mathrm{2}} \\ $$$$\Rightarrow\:\:{x}−\sqrt{\mathrm{2}}{a}^{\mathrm{3}} \:=\:\frac{\mathrm{12}}{{a}^{\mathrm{2}} } \\ $$$${x}\:=\:\sqrt{\mathrm{2}}{a}^{\mathrm{3}} +\frac{\mathrm{12}}{{a}^{\mathrm{2}} } \\ $$$${which}\:{has}\:{a}\:{minimum}\:{value} \\ $$$${when}\:\:\:\:\mathrm{3}\sqrt{\mathrm{2}}{a}^{\mathrm{2}} \:=\:\frac{\mathrm{24}}{{a}^{\mathrm{3}} } \\ $$$$\Rightarrow\:\:\:\:\:{a}^{\mathrm{5}} \:=\:\left(\sqrt{\mathrm{2}}\right)^{\mathrm{5}} \:\:\:\Rightarrow\:\:{a}=\:\sqrt{\mathrm{2}}\: \\ $$$$\Rightarrow\:\:\:{x}_{{min}} \:=\:\mathrm{4}+\frac{\mathrm{12}}{\mathrm{2}}\:=\:\mathrm{10}\:. \\ $$
Commented by Joel578 last updated on 28/Sep/18
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much} \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 28/Sep/18
$${f}\left({a},{b}\right)=\sqrt{\mathrm{2}}\:{a}^{\mathrm{3}} +\mathrm{3}\left({ab}−{b}^{\mathrm{2}} \right)^{−\mathrm{1}} \\ $$$$\frac{\partial{f}}{\partial{a}}=\mathrm{3}\sqrt{\mathrm{2}}\:{a}^{\mathrm{2}} −\mathrm{3}\left({ab}−{b}^{\mathrm{2}} \right)^{−\mathrm{2}} \left({b}\right) \\ $$$$\frac{\partial{f}}{\partial{a}}=\mathrm{0}=\mathrm{3}\sqrt{\mathrm{2}}\:{a}^{\mathrm{2}} −\frac{\mathrm{3}{b}}{\left({ab}−{b}^{\mathrm{2}} \right)^{\mathrm{2}} }\:\:\:\:{we}\:{get}\:{a}=\sqrt{\mathrm{2}}\: \\ $$$$ \\ $$$$\frac{\partial^{\mathrm{2}} {f}}{\partial{a}^{\mathrm{2}} }=\mathrm{6}\sqrt{\mathrm{2}}\:{a}+\mathrm{6}\left({ab}−{b}^{\mathrm{2}} \right)^{−\mathrm{3}} \left({b}\right)^{\mathrm{2}} \\ $$$$\frac{\partial{f}}{\partial{b}}=\mathrm{0}−\mathrm{3}\left({ab}−{b}^{\mathrm{2}} \right)^{−\mathrm{2}} \left({a}−\mathrm{2}{b}\right) \\ $$$$\frac{\partial{f}}{\partial{b}}=\mathrm{0}=\frac{−\mathrm{3}}{\left({ab}−{b}^{\mathrm{2}} \right)^{\mathrm{2}} }\left({a}−\mathrm{2}{b}\right)\:{we}\:{get}\:\:{b}=\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}\:} \\ $$$${a}=\mathrm{2}{b} \\ $$$$ \\ $$$$\mathrm{3}\sqrt{\mathrm{2}}{a}^{\mathrm{2}} −\frac{\mathrm{3}{b}}{\left({ab}−{b}^{\mathrm{2}} \right)^{\mathrm{2}} }=\mathrm{0} \\ $$$$\mathrm{12}\sqrt{\mathrm{2}}\:{b}^{\mathrm{2}} −\frac{\mathrm{3}{b}}{{b}^{\mathrm{4}} }=\mathrm{0} \\ $$$$\mathrm{12}\sqrt{\mathrm{2}}\:{b}^{\mathrm{6}} −\mathrm{3}{b}=\mathrm{0} \\ $$$$\mathrm{3}{b}\left(\mathrm{4}\sqrt{\mathrm{2}}\:{b}^{\mathrm{5}} −\mathrm{1}\right)=\mathrm{0} \\ $$$${b}=\left(\frac{\mathrm{1}}{\mathrm{4}\sqrt{\mathrm{2}}}\right)^{\frac{\mathrm{1}}{\mathrm{5}}} =\left\{\left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:\right)^{\mathrm{5}} \right\}^{\frac{\mathrm{1}}{\mathrm{5}}} =\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}} \\ $$$${a}=\mathrm{2}×\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}=\sqrt{\mathrm{2}}\: \\ $$$$\frac{\partial^{\mathrm{2}} {b}}{\partial{b}^{\mathrm{2}} }=\mathrm{6}\left({ab}−{b}^{\mathrm{2}} \right)^{−\mathrm{3}} \left({a}−\mathrm{2}{b}\right)^{\mathrm{2}} −\mathrm{3}\left({ab}−{b}^{\mathrm{2}} \right)\left(\mathrm{0}−\mathrm{2}\right) \\ $$$${now}\:{putting}\:{the}\:{value}\:{in}\:\frac{\partial^{\mathrm{2}} {f}}{\partial{a}^{\mathrm{2}} }\:{and}\:\frac{\partial^{\mathrm{2}} {f}}{\partial{b}^{\mathrm{2}} } \\ $$$$\frac{\partial^{\mathrm{2}} {f}}{\partial{a}^{\mathrm{2}} }=\mathrm{6}\sqrt{\mathrm{2}}\:{a}+\frac{\mathrm{6}{b}^{\mathrm{2}} }{\left({ab}−{b}^{\mathrm{2}} \right)^{\mathrm{3}} } \\ $$$$=\mathrm{6}\sqrt{\mathrm{2}}\:×\sqrt{\mathrm{2}}\:+\frac{\mathrm{6}×\frac{\mathrm{1}}{\mathrm{2}}}{\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{3}} }=\mathrm{12}+\frac{\mathrm{3}}{\frac{\mathrm{1}}{\mathrm{8}}}=\mathrm{36} \\ $$$$\frac{\partial^{\mathrm{2}} {f}}{\partial{b}^{\mathrm{2}} }=\mathrm{6}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\right)^{−\mathrm{3}} \left(\sqrt{\mathrm{2}}\:−\frac{\mathrm{2}}{\:\sqrt{\mathrm{2}}\:}\right)^{\mathrm{2}} +\mathrm{6}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\right)=\mathrm{3} \\ $$$$\frac{\partial}{\partial{a}}\left(\frac{\partial{f}}{\partial{b}}\right)=\frac{\partial}{\partial{a}}\left\{\frac{\mathrm{6}{b}−\mathrm{3}{a}}{\left({ab}−{b}^{\mathrm{2}} \right)^{\mathrm{2}} }\right\} \\ $$$$\:\:=\frac{\left({ab}−{b}^{\mathrm{2}} \right)^{\mathrm{2}} \left(−\mathrm{3}\right)−\left(\mathrm{6}{b}−\mathrm{3}{a}\right)×\mathrm{2}\left({ab}−{b}^{\mathrm{2}} \right)\left({b}\right)}{\left({ab}−{b}^{\mathrm{2}} \right)^{\mathrm{4}} } \\ $$$$=\frac{\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} \left(−\mathrm{3}\right)−\left(\frac{\mathrm{6}}{\:\sqrt{\mathrm{2}}}−\mathrm{3}\sqrt{\mathrm{2}}\:\right)×\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\right)\left(\frac{\mathrm{1}}{\mathrm{2}}\right)}{\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{4}} } \\ $$$$=\frac{\frac{−\mathrm{3}}{\mathrm{4}}}{\frac{\mathrm{1}}{\mathrm{16}}}=−\mathrm{12} \\ $$$$\frac{\partial^{\mathrm{2}} {f}}{\partial{a}^{\mathrm{2}} }×\frac{\partial^{\mathrm{2}} {f}}{\partial{b}^{\mathrm{2}} }−\left\{\frac{\partial}{\partial{a}}\left(\frac{\partial{f}}{\partial{b}}\right)\right\}^{\mathrm{2}} \\ $$$$=\mathrm{36}×\mathrm{3}−\left(−\mathrm{12}\right)^{\mathrm{2}} \\ $$$$=\mathrm{108}−\mathrm{144}=−\mathrm{36}<\mathrm{0}\:\:\boldsymbol{{saddle}}\:\boldsymbol{{point}}… \\ $$$$ \\ $$