First slide

Multiple and sub- multiple Angles

Question

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

Moderate

A

$[7 / 8, 5 / 4]$
B

$[1 / 2, 5 / 8]$
C

$[1 / 4, 1]$
D

none of these

Solution

We have,
$\begin{array}{l} k = \sin^{6} x + \cos^{6} x \\ \Rightarrow k = (\sin^{2} x + \cos^{2} x) (\sin^{4} x + \cos^{4} x - \sin^{2} x \cos^{2} x) \\ \Rightarrow k = (1 - 3 \sin^{2} x \cos^{2} x) \\ \Rightarrow k = (1 - \frac{3}{4} \sin^{2} 2 x) \end{array}$
Now,
$0 \leq \frac{3}{4} \sin^{2} 2 x \leq \frac{3}{4},$ for all $x$
$\begin{array}{l} \Rightarrow - \frac{3}{4} \leq - \frac{3}{4} \sin^{2} 2 x \leq 0, for all x \\ \Rightarrow 1 - \frac{3}{4} \leq 1 - \frac{3}{4} \sin^{2} 2 x \leq 1, for all x \\ \Rightarrow \frac{1}{4} \leq 1 - \frac{3}{4} \sin^{2} 2 x \leq 1, for all x \\ \Rightarrow \frac{1}{4} \leq k \leq 1 \end{array}$

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