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. $[0,\frac{\pi }{4}]$
• D. $[\frac{\pi }{4},\frac{\pi }{2}]$

Answer 1

The correct option is D $[\frac{\pi }{4},\frac{\pi }{2}]$

$f\left(x\right)={\sin }^{4}x+{\cos }^{4}x{f}^{\prime }\left(x\right)=4{\sin }^{3}x\cos x-4{\cos }^{3}x\sin x=4\sin x\cos x({\sin }^{2}x-{\cos }^{2}x)=-2\sin 2x.\cos 2x=-\sin 4x{f}^{\prime }\left(x\right)\ge 0\text{(for increasing function)}\Rightarrow \sin 4x\le 0\Rightarrow \pi \le 4x\le 2\pi$
$x\in [\frac{\pi }{4},\frac{\pi }{2}]$