Question Number 16090 by Tinkutara last updated on 17/Jun/17
$$\mathrm{The}\:\mathrm{maximum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{expression} \\ $$$$\mid\sqrt{\mathrm{sin}^{\mathrm{2}} \:{x}\:+\:\mathrm{2}{a}^{\mathrm{2}} }\:−\:\sqrt{\mathrm{2}{a}^{\mathrm{2}} \:−\:\mathrm{3}\:−\:\mathrm{cos}^{\mathrm{2}} \:{x}}\mid; \\ $$$$\mathrm{where}\:'{a}'\:\mathrm{and}\:'{x}'\:\mathrm{are}\:\mathrm{real}\:\mathrm{numbers},\:\mathrm{is} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{4} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{2} \\ $$$$\left(\mathrm{3}\right)\:\sqrt{\mathrm{2}} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{0} \\ $$
Answered by ajfour last updated on 18/Jun/17
$${for}\:{maximum}\:{value}\:{let}\:{sinx}=\mathrm{1}, \\ $$$$\mathrm{cos}\:{x}=\mathrm{0} \\ $$$${then}\:{maximum}\:{of} \\ $$$$\:{f}\left({z}\right)=\sqrt{\mathrm{1}+\mathrm{2}{z}^{\mathrm{2}} }−\sqrt{\mathrm{2}{z}^{\mathrm{2}} −\mathrm{3}} \\ $$$$\:{is}\:{an}\:{even}\:{function}\:{with} \\ $$$$\:{f}\:'\left({z}\right)<\mathrm{0}\:{for}\:{z}>\sqrt{\mathrm{3}/\mathrm{2}} \\ $$$$\:{hence}\:{maximum}\:{when}\:\mathrm{2}{z}^{\mathrm{2}} =\mathrm{3} \\ $$$$\:\:\:\:{and}\:{maximum}\:{value}\:{of} \\ $$$${expression}\:=\:\mathrm{2}\:. \\ $$
Commented by Tinkutara last updated on 18/Jun/17
$$\mathrm{Why}\:{f}\left({z}\right)\:\mathrm{is}\:\mathrm{an}\:\mathrm{even}\:\mathrm{function}\:\mathrm{with} \\ $$$${f}\:'\left({z}\right)\:<\:\mathrm{0}\:? \\ $$