Question

Let $f\left(x\right)={\sin }^{4}x+{\cos }^{4}x$. Then f is an increasing function in the interval:

• A. $[\frac{5\pi }{8},\frac{3\pi }{4}]$
• B. $[\frac{\pi }{2},\frac{5\pi }{8}]$
• C. $[\frac{\pi }{4},\frac{\pi }{2}]$
• D. $[0,\frac{\pi }{4}]$

Answer 1

Answer: (C)

$[\frac{\pi }{4},\frac{\pi }{2}]$

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

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

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

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

$=-\sin 4x$

$f\left(x\right)$ is increasing when $f^{\prime }\left(x\right)>0$

$\Rightarrow -\sin 4x>0$

$\Rightarrow \sin 4x<0$

$\Rightarrow 4x\in (\pi ,2\pi )$

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