Question

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

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

Answer 1

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

$\begin{array}{cc}& f\left(x\right)={\sin }^{4}x+{\cos }^{4}x\\ & f^{\prime }\left(x\right)=4{\sin }^{3}x\left(\cos x\right)+4{\cos }^{3}x(-\sin x)\\ & =4\sin x\cos x({\sin }^{2}x-{\cos }^{2}x)\\ & =-\sin 4x\\ & \Rightarrow f\left(x\right)\text{is}\text{increasing when}{f}^{\prime }\left(x\right)>0\\ & \Rightarrow \sin 4x<0\Rightarrow x\in (\frac{\pi }{4},\frac{\pi }{2})\\ \therefore & \text{option}B\text{is correct.}\end{array}$