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$The range of \cos x + 4 \sqrt{2} \sin (x - \frac{π}{4}) + 6 is$

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a

[1, 11]

b

[- 1, 11]

c

[2, 10]

d

[10, 11]

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detailed solution

Correct option is A

$f (x) = \cos x + 4 \sqrt{2} \sin (x - \frac{π}{4}) + 6 f (x) = \cos x + 4 \sqrt{2} (\sin x \cos \frac{π}{4} - \cos x \sin \frac{π}{4}) + 6 \Rightarrow f (x) = \cos x + 4 \sqrt{2} (\frac{\sin x - \cos x}{\sqrt{2}}) + 6 \Rightarrow f (x) = \cos x + 4 (\sin x - \cos x) + 6 \Rightarrow f (x) = 4 \sin x - 3 \cos x + 6 R a n g e = [6 - \sqrt{16 + 9}, 6 + \sqrt{16 + 9}] = [6 - 5, 6 + 5] = [1, 11]$

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