MCQ
If $K = sin^6x + cos^6x$, then $K$ belongs to the interval
- A$\left[ {\frac{7}{8},\frac{5}{4}} \right]$
- B$\left[ {\frac{1}{5},\frac{5}{8}} \right]$
- ✓$\left[ {\frac{1}{4},1} \right]$
- DNone of these
$=(1)^{3}-3 \sin ^{2} x \cos ^{2} x(1)$
$=1-\frac{3}{4} \sin ^{2} 2 x$
Now, $0 \leq \sin ^{2} 2 x \leq 1 \quad \Rightarrow \quad 0 \leq \frac{3}{4} \sin ^{2} 2 x \leq \frac{3}{4}$
$\Rightarrow \quad-\frac{3}{4} \leq-\frac{3}{4} \sin ^{2} 2 x \leq 0$
$\Rightarrow \quad 1-\frac{3}{4} \leq 1-\frac{3}{4} \sin ^{2} 2 x \leq 1$
$\Rightarrow \quad \frac{1}{4} \leq 1-\frac{3}{4} \sin ^{2} 2 x \leq 1$
$\Rightarrow \quad \frac{1}{4} \leq \mathrm{K} \leq 1 \Rightarrow \mathrm{K} \in\left[\frac{1}{4}, 1\right]$
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