Question

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

• A. $0<x<\frac{\pi }{8}$
• B. $\frac{\pi }{4}<x<\frac{3\pi }{8}$
• C. $\frac{3\pi }{8}<x<\frac{5\pi }{8}$
• D. $\frac{5\pi }{4}<x<\frac{3\pi }{4}$

Answer 1

The correct option is B $\frac{\pi }{4}<x<\frac{3\pi }{8}$

$f\left(x\right)={\sin }^{4}x+{\cos }^{4}x$

$f\left(x\right)=4{\sin }^{3}x\cos x-4{\cos }^{3}x\sin x$

$=4\sin x\cos x({\sin }^{2}x-{\cos }^{2}x)$

$=-4\sin x\cos x\cos 2x$

$=-2\sin 2x\cos 2x$

$=-{\sin }^{4}x$

$f=$ increasing $\Rightarrow f\left(x\right)>0$

$\Rightarrow -\sin 4x>0$

$\Rightarrow \sin 4x<0$

$\Rightarrow 4x\in (\frac{\pi }{2},\frac{3\pi }{2})$

$x\in (\frac{\pi }{8},\frac{3\pi }{8})$

$\therefore \left(B\right)$ option is correct