Question

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

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

Answer 1

Answer: (B)

$f(x)=\sin ^{4} x+\cos ^{4} x$
$f'(x)=4 \sin ^{3} x \cos x-4 \cos ^{3} x \sin x$
$=4 \sin x \cos x\left(\sin ^{2} x-\cos ^{2} x\right)$
$=-2 \sin 2 x . \cos 2 x$
$=-\sin 4 x>0$
$\Rightarrow \sin 4 x<0$
$\Rightarrow \pi< 4 x< 2 \pi$
$\frac{\pi}{4}< x< \frac{\pi}{2}$