Question

The number of solutions of ${\sin }^{7}x+{\cos }^{7}x=1,x\in [0,4\pi ]$ is equal to

• A. 11
• B. 7
• C. 5
• D. 9

Answer 1

Answer: (C)

${\sin }^{7}x\le {\sin }^{2}x\le \lambda 1$
and ${\cos }^{7}x\le {\cos }^{2}x\le 1$
also ${\sin }^{2}x+{\cos }^{2}x=1$
$\Rightarrow$ equality must hold for (1) $\&(2)$
$\Rightarrow {\sin }^{7}x={\sin }^{2}x\&{\cos }^7={\cos }^{2}x$
$\Rightarrow \sin x=0\&\cos x=1$
or
$\cos x=0\&\sin x=1$
$\Rightarrow x=0,2\pi ,4\pi ,\frac{\pi }{2},\frac{5\pi }{2}$
$\Rightarrow 5$ solutions