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If $k = \sin^{6} x + \cos^{6} x,$ then $k$ belongs to the interval

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a

[7/8,5/4]

b

[1/2,5/8]

c

[1/4,1]

d

none of these

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detailed solution

Correct option is C

We have,k=sin6⁡x+cos6⁡x⇒ k=sin2⁡x+cos2⁡xsin4⁡x+cos4⁡x−sin2⁡xcos2⁡x⇒ k=1−3sin2⁡xcos2⁡x⇒ k=1−34sin2⁡2xNow,0≤34sin2⁡2x≤34, for all x⇒ −34≤−34sin2⁡2x≤0, for all x⇒ 1−34≤1−34sin2⁡2x≤1, for all x⇒ 14≤1−34sin2⁡2x≤1, for all x⇒ 14≤k≤1

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