The range of $1 + 12 {sin}^{2} x {cos}^{2} x$ is

[- 1, 1]

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[- 2, 1]

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[1, 2]

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[1, 4]

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Solution

The correct option is C $[1, 4]$
Let $f (x) = 1 + 12 {sin}^{2} x {cos}^{2} x$
$= 1 + 3 (2 sin x cos x)^{2}$
$f (x) = 1 + 3 (sin 2 x)^{2} [∵ sin 2 x = 2 sin x cos x]$
$∴$ Max. value of $f (x) = 1 + 3 (1)^{2} = 4$ , when $sin 2 x = 1.$
Min. value of $f (x) = 1 + 3 (0) = 1$ , when $sin 2 x = 0$
So, range of $f (x) \in [1, 4]$

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The range of 1+12sin2xcos2x is

The correct option is C [1,4]Let f(x)=1+12sin2xcos2x=1+3(2sinxcosx)2f(x)=1+3(sin2x)2[∵sin2x=2sinxcosx]∴ Max. value of f(x)=1+3(1)2=4, when sin2x=1.Min. value of f(x)=1+3(0)=1, when sin2x=0So, range of f(x)∈[1,4]

The range of $1 + 12 {sin}^{2} x {cos}^{2} x$ is