First slide

Monotonicity

Question

The function $f (x) = \sin^{4} x + \cos^{4} x$ increases if

Easy

A

$o < x < \frac{π}{8}$
B

$\frac{π}{4} < x < \frac{π}{2}$
C

$\frac{3 π}{8} < x < \frac{5 π}{8}$
D

$\frac{5 π}{8} < x < \frac{3 π}{8}$

Solution

$\begin{array}{l} f (x) = \sin^{4} x + \cos^{4} x \\ = {(\cos^{2} x + \sin^{2} x)}^{2} - 2 \sin^{2} x \cos^{2} x \\ = 1 - \frac{1}{2} \sin^{2} 2 x \\ = 1 - \frac{1}{4} (1 - \cos 4 x) = \frac{3}{4} + \frac{1}{4} \cos x \\ ∴ f^{1} (x) = - \sin 4 x \\ f o r f (x) t o b e i n c r e a s i n g f^{1} (x) > 0 \\ \Rightarrow \sin 4 x < 0 \\ ⇒ π < 4 x < 2 π \\ \Rightarrow \frac{π}{4} < x < \frac{π}{2} \end{array}$

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