The correct option is D [1, 11] f(x)=cos x+4√(2) sin (x π4)+6 =cosx+4√(2)[sinx.cos(π/4) sin(π/4).cosx]+6 =cos x+4 (sin x cos x)+6 ........ ...

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Composite Function

cos x+4√2sinx...

Question

$cos x + 4 \sqrt{2} s i n (x - \frac{π}{4}) + 6 \in$

[1, 11]

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[-1, 11]

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[2, 10]

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[11, 1]

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Solution

The correct option is D [1, 11]
$f (x) = c o s x + 4 \sqrt{2} s i n (x - \frac{π}{4}) + 6$

$= c o s x + 4 \sqrt{2} [s i n x . c o s (π / 4) - s i n (π / 4) . c o s x] + 6$

$= c o s x + 4 (s i n x - c o s x) + 6$ $. . . . . . . . . . . [s i n (π / 4) = c o s (π / 4) = 1 / \sqrt{2}]$
$= 4 s i n x - 3 c o s x + 6$
Range of $4 s i n x - 3 c o s x ϵ [- \sqrt{4^{2} + 3^{2}}, \sqrt{4^{2} + 3^{2}}]$
$ϵ [- 5, 5]$
So, range of $f (x) ϵ [- 5 + 6, 5 + 6]$
$\Rightarrow ϵ [1, 11]$

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