Question

The following function are decreasing in given interval
$f\left(x\right)={\sin }^{4}x+{\cos }^{4}x,x\in (0,\pi /4)$

• A. True
• B. False

Answer 1

The correct option is B False

$f\left(x\right)={\sin }^{4}x+{\cos }^{4}x=\frac{1}{4}\left(\right(2{\sin }^{2}x{)}^2+\left(2{\cos }^{2}x{)}^2\right)=\frac{1}{4}\left(\right(1-\cos 2x{)}^2+(1+\cos 2x{)}^2)=\frac{1}{4}(2+2{\cos }^{2}2x)=\frac{3}{4}+\frac{1}{4}\cos 4x$

As $\cos \theta$ decreases in the interval $\theta \in (\frac{\pi }{2},\pi )$

$\cos 4x$ decreases in the interval $4x\in (\frac{\pi }{2},\pi )⟹x\in (\frac{\pi }{8},\frac{\pi }{4})$

So $f\left(x\right)$ decreases in the interval $x\in (\frac{\pi }{8},\frac{\pi }{4})$