Question Number 171593 by cortano1 last updated on 18/Jun/22
$$\:{The}\:{maximum}\:{value}\:{of}\:{the} \\ $$$${expression}\:\mid\sqrt{\mathrm{sin}\:^{\mathrm{2}} {x}+\mathrm{2}{a}^{\mathrm{2}} }\:−\sqrt{\mathrm{2}{a}^{\mathrm{2}} −\mathrm{1}−\mathrm{cos}\:^{\mathrm{2}} {x}}\:\mid\: \\ $$$${where}\:{a}\:{and}\:{x}\:{real}\:{numbers}\:{is}−−− \\ $$
Commented by infinityaction last updated on 18/Jun/22
$$\:\:\:{for}\:{maximum}\:{value} \\ $$$$\:\:\:\:\mathrm{2}{a}^{\mathrm{2}} −\mathrm{1}−\mathrm{cos}\:^{\mathrm{2}} {x}\:=\:\mathrm{0} \\ $$$$\:\:\:\:\:\mathrm{cos}\:^{\mathrm{2}} \:=\:\mathrm{2}{a}^{\mathrm{2}} −\mathrm{1} \\ $$$$\:\:\:\:\:\mathrm{1}−\mathrm{sin}\:^{\mathrm{2}} {x}\:=\:\mathrm{2}{a}^{\mathrm{2}} −\mathrm{1} \\ $$$$\:\:\:\:\:\mathrm{sin}\:^{\mathrm{2}} {x}+\mathrm{2}{a}^{\mathrm{2}} \:=\:\mathrm{2} \\ $$$$\:\:\:{so} \\ $$$$\:\:\:\:\:\mid\sqrt{\mathrm{2}}−\mathrm{0}\mid\:=\:\sqrt{\mathrm{2}}\:\left({maximum}\:{value}\right) \\ $$$$ \\ $$
Commented by cortano1 last updated on 18/Jun/22
$${yes}... \\ $$