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  4. If g ( x )=2 √2 sin x + cos x then maximum value of P ( x )=√ g ( x )-2+√4- g ( x ) is

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Q. If $g ( x )=2 \sqrt{2} \sin x +\cos x$ then maximum value of $P ( x )=\sqrt{ g ( x )-2}+\sqrt{4- g ( x )}$ is

Relations and Functions - Part 2

Solution:

$g(x)=2 \sqrt{2} \sin x+\cos x $
$g(x) \in[-3,3] $
$y=\sqrt{g(x)-2}+\sqrt{4-g(x)} $
$y^2=g(x)-2+4-g(x)+2 \sqrt{g(x)-2} \sqrt{4-g(x)}$
$y^2=2+2 \sqrt{4 g-g^2-8+2 g}=2+2 \sqrt{-[g(x)]^2+6(g(x))-8}=2+2 \sqrt{1-[g(x)-3]^2}$
$\text { where } g(x)=3 \text { then } y^2 \max ^2=4$
$y_{\max }=2$