Question Number 6257 by 314159 last updated on 20/Jun/16
$${Given}\:{that}\:{a}\:{and}\:{b}\:{are}\:{positive}\:{real}\:{number} \\ $$$${such}\:{that}\:{b}<\mathrm{4}{a}+\mathrm{1},{show}\:{that}\:\frac{\mathrm{2}{a}+{b}}{\mathrm{4}{a}+\mathrm{1}}<\sqrt{\mathrm{4}{a}^{\mathrm{2}} +{b}}\:. \\ $$
Answered by Yozzii last updated on 20/Jun/16
$$\frac{\mathrm{2}{a}+{b}}{\mathrm{4}{a}+\mathrm{1}}<\sqrt{\mathrm{4}{a}^{\mathrm{2}} +{b}}\:\:\:\:\:\:{a},{b}>\mathrm{0} \\ $$$$\Rightarrow\frac{\left(\mathrm{2}{a}+{b}\right)^{\mathrm{2}} }{\left(\mathrm{4}{a}+\mathrm{1}\right)^{\mathrm{2}} }<\mathrm{4}{a}^{\mathrm{2}} +{b} \\ $$$$−−−−−−−−−−−−−−−−−−−−−−−−−−− \\ $$$${Let}\:\phi=\frac{\left(\mathrm{2}{a}+{b}\right)^{\mathrm{2}} }{\left(\mathrm{4}{a}+\mathrm{1}\right)^{\mathrm{2}} }−\mathrm{4}{a}^{\mathrm{2}} −{b}. \\ $$$$\phi=\frac{\mathrm{4}{a}^{\mathrm{2}} +\mathrm{4}{ab}+{b}^{\mathrm{2}} −\left(\mathrm{16}{a}^{\mathrm{2}} +\mathrm{8}{a}+\mathrm{1}\right)\left(\mathrm{4}{a}^{\mathrm{2}} +{b}\right)}{\left(\mathrm{4}{a}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\phi=\frac{\mathrm{4}{a}^{\mathrm{2}} +\mathrm{4}{ab}+{b}^{\mathrm{2}} −\mathrm{64}{a}^{\mathrm{4}} −\mathrm{16}{a}^{\mathrm{2}} {b}−\mathrm{32}{a}^{\mathrm{3}} −\mathrm{8}{ab}−\mathrm{4}{a}^{\mathrm{2}} −{b}}{\left(\mathrm{4}{a}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\phi=\frac{−\mathrm{64}{a}^{\mathrm{4}} −\mathrm{16}{a}^{\mathrm{2}} {b}−\mathrm{32}{a}^{\mathrm{3}} −\mathrm{4}{ab}+{b}\left({b}−\mathrm{1}\right)}{\left(\mathrm{4}{a}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$$${Since}\:{b}<\mathrm{4}{a}+\mathrm{1}\Rightarrow{b}−\mathrm{1}<\mathrm{4}{a}\Rightarrow{b}\left({b}−\mathrm{1}\right)<\mathrm{4}{ab} \\ $$$$\Rightarrow\phi<\frac{−\mathrm{64}{a}^{\mathrm{4}} −\mathrm{16}{a}^{\mathrm{2}} {b}−\mathrm{32}{a}^{\mathrm{3}} −\mathrm{4}{ab}+\mathrm{4}{ab}}{\left(\mathrm{4}{a}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\phi<\frac{−\mathrm{16}{a}^{\mathrm{2}} \left(\mathrm{4}{a}^{\mathrm{2}} +{b}+\mathrm{2}{a}\right)}{\left(\mathrm{4}{a}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$$${Since}\:{a},{b}>\mathrm{0}\Rightarrow\:\mathrm{4}{a}^{\mathrm{2}} +\mathrm{2}{a}+{b}>\mathrm{0},\:−\mathrm{16}{a}^{\mathrm{2}} <\mathrm{0} \\ $$$${and}\:\left(\mathrm{4}{a}+\mathrm{1}\right)^{\mathrm{2}} >\mathrm{0}.\:{Thus},\:\phi<\mathrm{0}. \\ $$$$\Rightarrow\mathrm{0}<\frac{\left(\mathrm{2}{a}+{b}\right)^{\mathrm{2}} }{\left(\mathrm{4}{a}+\mathrm{1}\right)^{\mathrm{2}} }<\mathrm{4}{a}^{\mathrm{2}} +{b} \\ $$$$\therefore\:\sqrt{\left(\frac{\mathrm{2}{a}+{b}}{\mathrm{4}{a}+\mathrm{1}}\right)^{\mathrm{2}} }<\sqrt{\mathrm{4}{a}^{\mathrm{2}} +{b}} \\ $$$$\Rightarrow\mid\frac{\mathrm{2}{a}+{b}}{\mathrm{4}{a}+\mathrm{1}}\mid<\sqrt{\mathrm{4}{a}^{\mathrm{2}} +{b}} \\ $$$${a},{b}>\mathrm{0}\Rightarrow\:\mathrm{2}{a}+{b}>\mathrm{0}\:{and}\:\mathrm{4}{a}+\mathrm{1}>\mathrm{0}. \\ $$$$\therefore\:\frac{\mathrm{2}{a}+{b}}{\mathrm{4}{a}+\mathrm{1}}<\sqrt{\mathrm{4}{a}^{\mathrm{2}} +{b}}. \\ $$