Question

Let $f\left(x\right)=(pq-q^2-2)x+{\int }_0^x({\sin }^4\theta +{\cos }^4\theta )d\theta$ is an decreasing function of $x\forall R$ $q$ being any real number independent of $x$ if

• A. $|p|\le 2$
• B. $p\in (0,2)$
• C. $p\in (-2,2)$
• D. $p\in (2,3)$

Answer 1

Answer: (A), (C)

The correct options are
A $|p|\le 2$
C $p\in (-2,2)$

$f^1\left(x\right)=(pq-q^2-2)+si{n}^{4}x+co{s}^{4}x$
$=(pq-q^2-2)+(si{n}^{2}x+co{s}^{2}x{)}^2-2si{n}^{2}xco{s}^{2}x$
$f^1\left(x\right)=pq-q^2-1-\frac{si{n}^{2}2x}{2}$
$f^1\left(x\right)<0$
$q^2-pq+1+\frac{si{n}^{2}2x}{2}>0$
$p^2-4(1+\frac{si{n}^{2}2x}{2})<0$
$p^2-4+2si{n}^{2}2x<0$
$p^2<4+2si{n}^{2}2x$
$pϵ(-2,2)$