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Question Number 16090 by Tinkutara last updated on 17/Jun/17

The maximum value of the expression  ∣(√(sin^2  x + 2a^2 )) − (√(2a^2  − 3 − cos^2  x))∣;  where ′a′ and ′x′ are real numbers, is  (1) 4  (2) 2  (3) (√2)  (4) 0

$$\mathrm{The}\:\mathrm{maximum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{expression} \\ $$$$\mid\sqrt{\mathrm{sin}^{\mathrm{2}} \:{x}\:+\:\mathrm{2}{a}^{\mathrm{2}} }\:−\:\sqrt{\mathrm{2}{a}^{\mathrm{2}} \:−\:\mathrm{3}\:−\:\mathrm{cos}^{\mathrm{2}} \:{x}}\mid; \\ $$$$\mathrm{where}\:'{a}'\:\mathrm{and}\:'{x}'\:\mathrm{are}\:\mathrm{real}\:\mathrm{numbers},\:\mathrm{is} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{4} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{2} \\ $$$$\left(\mathrm{3}\right)\:\sqrt{\mathrm{2}} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{0} \\ $$

Answered by ajfour last updated on 18/Jun/17

for maximum value let sinx=1,  cos x=0  then maximum of   f(z)=(√(1+2z^2 ))−(√(2z^2 −3))   is an even function with   f ′(z)<0 for z>(√(3/2))   hence maximum when 2z^2 =3      and maximum value of  expression = 2 .

$${for}\:{maximum}\:{value}\:{let}\:{sinx}=\mathrm{1}, \\ $$$$\mathrm{cos}\:{x}=\mathrm{0} \\ $$$${then}\:{maximum}\:{of} \\ $$$$\:{f}\left({z}\right)=\sqrt{\mathrm{1}+\mathrm{2}{z}^{\mathrm{2}} }−\sqrt{\mathrm{2}{z}^{\mathrm{2}} −\mathrm{3}} \\ $$$$\:{is}\:{an}\:{even}\:{function}\:{with} \\ $$$$\:{f}\:'\left({z}\right)<\mathrm{0}\:{for}\:{z}>\sqrt{\mathrm{3}/\mathrm{2}} \\ $$$$\:{hence}\:{maximum}\:{when}\:\mathrm{2}{z}^{\mathrm{2}} =\mathrm{3} \\ $$$$\:\:\:\:{and}\:{maximum}\:{value}\:{of} \\ $$$${expression}\:=\:\mathrm{2}\:. \\ $$

Commented by Tinkutara last updated on 18/Jun/17

Why f(z) is an even function with  f ′(z) < 0 ?

$$\mathrm{Why}\:{f}\left({z}\right)\:\mathrm{is}\:\mathrm{an}\:\mathrm{even}\:\mathrm{function}\:\mathrm{with} \\ $$$${f}\:'\left({z}\right)\:<\:\mathrm{0}\:? \\ $$

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