Question

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

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

Answer 1

The correct option is A $0<x<\pi /4$

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

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

$f^{\prime }\left(x\right)=4\sin x\cos x({\sin }^{2}x-{\cos }^{2}x)=0$

$\Rightarrow \sin x\cos x=0,{\sin }^{2}x-{\cos }^{2}x=0$

$\sin x=0,\to x=0$

$\cos x=0,\to x=\frac{\pi }{2}$

${\sin }^{2}x-{\cos }^{2}x=0\to x=\frac{\pi }{4}$

$[\frac{\pi }{4},\frac{\pi }{2}],f^{\prime }\left(\frac{\pi }{3}\right)\to +\Rightarrow f\left(x\right)$ is increasing.