Question
The function $$f\left( x \right) = {\sin ^4}x + {\cos ^4}x$$ increases if :
A.
$$0 < x < \frac{\pi }{8}$$
B.
$$\frac{{\pi }}{4} < x < \frac{{3\pi }}{8}$$
C.
$$\frac{{3\pi }}{4} < x < \frac{{5\pi }}{8}$$
D.
$$\frac{{5\pi }}{8} < x < \frac{{3\pi }}{4}$$
Answer :
$$\frac{{\pi }}{4} < x < \frac{{3\pi }}{8}$$
Solution :
$$\eqalign{
& f'\left( x \right) > 0 \cr
& \Rightarrow 4{\sin ^3}x.\cos \,x - 4{\cos ^3}x.\sin \,x > 0 \cr
& \Rightarrow \sin \,x.\cos \,x\left( {{{\sin }^2}x - {{\cos }^2}x} \right) > 0 \cr
& {\text{or }}\sin \,2x.\cos \,2x < 0\,\,\,\,\,\,{\text{or }}\sin \,4x < 0 \cr
& \therefore \pi < 4x < 2\pi \,\,\,\,{\text{or }}3\pi < 4x < 4\pi \cr
& \therefore \frac{\pi }{4} < x < \frac{\pi }{2}\,\,\,{\text{or }}\frac{{3\pi }}{4} < x < \pi \cr} $$