Question Number 167278 by mnjuly1970 last updated on 11/Mar/22
Answered by mr W last updated on 11/Mar/22
$${f}\left({x}\right)=\frac{\sqrt{{x}\left({x}^{\mathrm{2}} +\mathrm{1}\right)}}{{x}^{\mathrm{2}} +\mathrm{1}+{x}}=\frac{\mathrm{1}}{\:\sqrt{\frac{{x}^{\mathrm{2}} +\mathrm{1}}{{x}}}+\sqrt{\frac{{x}}{{x}^{\mathrm{2}} +\mathrm{1}}}} \\ $$$$=\frac{\mathrm{1}}{\:\sqrt{{x}+\frac{\mathrm{1}}{{x}}}+\frac{\mathrm{1}}{\:\sqrt{{x}+\frac{\mathrm{1}}{{x}}}}} \\ $$$$\leqslant\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}}=\frac{\sqrt{\mathrm{2}}}{\mathrm{3}}={f}_{{max}} \\ $$
Commented by mnjuly1970 last updated on 12/Mar/22
$${thanks}\:{alot}\:\:{Sir}\:\:{W} \\ $$