The correct option is D [0,3] Let f(x)=(sin x+cos x)^2+cos ^2 (π/4 x ) =1+2 sin x cos x+ [cos 2(π/4 x)+1/2 ] =3/2 +sin 2x+sin 2x/2 f( ...

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Algebra of Derivatives

sinx+cosx2+co...

Question

$(s i n x + c o s x)^{2} + {cos}^{2} (\frac{π}{4} - x) \in$

[0, 1]

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[0, 2]

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[1, 2]

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[0, 3]

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Solution

The correct option is D $[0, 3]$
Let $f (x) = (s i n x + c o s x)^{2} + c o s^{2} (\frac{π}{4} - x)$
$= 1 + 2 s i n x c o s x + [\frac{c o s 2 (\frac{π}{4} - x) + 1}{2}]$
$= \frac{3}{2} + s i n 2 x + \frac{s i n 2 x}{2}$
$f (x) = \frac{3}{2} + \frac{3}{2} s i n 2 x$
$∴$ Range of $f (x) ϵ [\frac{3}{2} - \frac{3}{2}, \frac{3}{2} + \frac{3}{2}]$
$\Rightarrow ϵ [0, 3]$

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