First slide

Extreme values of trigonometric functions in trigonometry

Question

If $f (x) = \sin^{6} x + \cos^{6} x$ then range of f (x) is

Moderate

A

$[\frac{1}{4}, 1]$
B

$[\frac{1}{4}, \frac{3}{4}]$
C

$[\frac{3}{4}, 1]$
D

none of these

Solution

$\begin{matrix} f (x) = \cos^{6} x + \sin^{6} x \\ = {(\cos^{2} x + \sin^{2} x)}^{3} - 3 \cos^{2} {xsin}^{2} x (\cos^{2} x + \sin^{2} x) \\ = 1 - 3 \cos^{2} x (1 - \cos^{2} x) \\ = 3 \cos^{4} x - 3 \cos^{2} x + 1 \\ = 3 (\cos^{4} x - \cos^{2} x + \frac{1}{3}) \\ = 3 ({(\cos^{2} x - \frac{1}{2})}^{2} + \frac{1}{12}) \end{matrix}$
Least value of $f (x) is \frac{1}{4}, when \cos^{2} x - \frac{1}{2} = 0$
Greatest value of $f (x) is 1, when \cos^{2} x = 0 or 1$

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