MCQ
The function $f (x) = sin^4x + cos^4x$ increases if :
- A$0 < x < \pi /8$
- ✓$\pi /4 < x < 3 \pi /8$
- C$3\pi/8 < x < 5\pi/8$
- D$5\pi/8 < x < 3\pi/4$
✓
Answer
Correct option: B.
$\pi /4 < x < 3 \pi /8$
b
$ f(x) =\sin ^{4} x+\cos ^{4} x $
$ 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 $
$f(x)$ increases if $f'\left( x \right) > 0$ i.e., $\sin 4 x<0$
$\Rightarrow \pi<4 \mathrm{x}<2 \pi \Rightarrow \pi / 4<\mathrm{x}<\pi / 2$
Since interval in choice $(2)$ in included in $(\pi / 4, \pi / 2).$
Hence the most appropriate answer is $(2)$
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