Explanation

(d)
$x^{12} - x^{9} + x^{4} - x + 1 > 0$, three cases arise
Case I When $x \leq 0$
$x^{12} > 0,\ - x^{9} > 0,\ x^{4} > 0,\ - x > 0$
$\Rightarrow \ \ x^{12} - x^{9} + x^{4} + x + 1 > 0,\ \forall\ x \leq 0$ ...(i)
Case II When $0 < x \leq 1$
$x^{9} < x^{4},\ x < 1\ \Rightarrow \ - x^{9} + x^{4} > 0$ and $1 - x > 0$
$\therefore\ \ x^{12} - x^{9} + x^{4} - x + 1 > 0,\ \forall\ 0 < x \leq 1$ ...(ii)
Case III When $x > 1$
$x^{12} > x^{9},\ x^{4} > x$
$\Rightarrow \ \ x^{12} - x^{9} + x^{4} - x + 1 > 0,\ \forall\ x > 1$ ...(iii)
$\therefore$ From Eqs. (i), (ii) and (iii) the above equation hold for $x \in R$