Let f ( x ) = ^4 ,x + ^4 ,x. Then f is an increasing function in … - Vidyadip

MCQ

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

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

✓

Answer

Correct option: C.

$\left[ {\frac{{\pi }}{4},\frac{{\pi }}{2}} \right]$

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

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

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

$=-2 \sin 2 x \cos 2 x$

$=-\sin 4 x$

$f(x)$ is increasing when $f(x)>0$

$\Rightarrow-\sin 4 x>0$

$\Rightarrow \quad \sin 4 x<0$

$\Rightarrow \quad x \in\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$

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