Question

The range of $x$ satisfying ${\sin }^4\left(\frac{x}{3}\right)+{\cos }^4\left(\frac{x}{3}\right)>\frac{1}{2}$ is (where $n\in Z$)

• A. $R-\left\{\right(2n+1)\frac{3\pi }{4},n\in Z\}$
• B. $R$
• C. $(2n+1)\frac{3\pi }{4},n\in Z$
• D. $(2n+1)\frac{3\pi }{2}$

Answer 1

The correct option is A $R-\left\{\right(2n+1)\frac{3\pi }{4},n\in Z\}$

${\sin }^4\left(\frac{x}{3}\right)+{\cos }^4\left(\frac{x}{3}\right)>\frac{1}{2}\Rightarrow 1-2{\sin }^2\left(\frac{x}{3}\right){\cos }^2\left(\frac{x}{3}\right)>\frac{1}{2}\Rightarrow 1-\frac{{\sin }^2\frac{2x}{3}}{2}>\frac{1}{2}\Rightarrow 2-{\sin }^2\frac{2x}{3}>1\Rightarrow {\sin }^2\frac{2x}{3}<1$
Which always true except when
${\sin }^2\frac{2x}{3}=1\Rightarrow \frac{2x}{3}=(2n+1)\frac{\pi }{2}\Rightarrow x=(2n+1)\frac{3\pi }{4}$

Hence, $x\in R-\left\{\right(2n+1)\frac{3\pi }{4},n\in Z\}$