Question

The inequality $\left|x^2\sin x+{\cos }^{2}x{e}^x+{\ln }^{2}x\right|<x^2\left|\sin x\right|+{\cos }^{2}x{e}^x+{\ln }^{2}x$ is true for $x\in$

• A. $\left(-\pi ,0\right)$
• B. $\left(0,\frac{\pi }{2}\right)$
• C. $\left(\frac{\pi }{2},\pi \right)$
• D. $\left[2n\pi ,\left(2n+1\right)\pi \right]\forall n\in N$

Answer 1

The correct option is A $\left(-\pi ,0\right)$

$\left|a+b+c\right|<\left|a\right|+\left|b\right|+\left|c\right|$
The above inequality is valid only if either one of $a,b$ or $c$ has different signs.
here, $e^x{\cos }^{2}x$ and ${\ln }^{2}x$ are both positive,
hence inequality will be valid only iff
$x^2\sin x<0$
$\Rightarrow \sin x<0$
$\Rightarrow x\in (-\pi ,0)$