Question Number 221168 by universe last updated on 25/May/25
Answered by Frix last updated on 26/May/25
$$\mathrm{Let}\:{b}={pa}\wedge{p}<\mathrm{0} \\ $$$$\lambda=\mathrm{min}\:\left(\frac{{p}−\mathrm{1}}{{ap}}\sqrt{\mathrm{2}{a}^{\mathrm{4}} {p}^{\mathrm{2}} +\mathrm{2}{a}^{\mathrm{2}} {p}+\mathrm{1}}\right) \\ $$$$\mathrm{Using}\:\mathrm{partial}\:\mathrm{differenciation}\:\mathrm{we}\:\mathrm{get} \\ $$$${a}=\frac{\mathrm{1}}{\:\sqrt{−\sqrt{\mathrm{2}}{p}}}\wedge{p}=−\mathrm{1} \\ $$$$\Rightarrow\:{a}=\frac{\mathrm{1}}{\:\sqrt[{\mathrm{4}}]{\mathrm{2}}}\wedge{b}=−\frac{\mathrm{1}}{\:\sqrt[{\mathrm{4}}]{\mathrm{2}}} \\ $$$$\lambda=\mathrm{2}\sqrt{\mathrm{2}\left(\sqrt{\mathrm{2}}−\mathrm{1}\right)} \\ $$$$\mathrm{3}\lambda^{\mathrm{2}} =\mathrm{24}\left(\sqrt{\mathrm{2}}−\mathrm{1}\right)\approx\mathrm{9}.\mathrm{94} \\ $$