Question

The largest interval for which $x^{12} - x^9 + x^4 - x + 1 > 0$ is

• A. $- 4 < x \le 0$
• B. 0 < x < 1
• C. - 100 < x < 100
• D. $- \infty < x < \infty$

Answer 1

Answer: (D)

$- \infty < x < \infty$

Answer 2

$x^{12} - x^9 + x^4 - x + 1 > 0$
Case I When $x \le 0 \Rightarrow x^{12} > 0, - x^9 > 0, x^4 > 0, - x > 0$
$\therefore x^{12} - x^9 + x^4 - x + 1 > 0, \, \forall \, x \le 0$ \hspace21mm ...(i)
Case II When 0 $< x \le 1$
$x^9 < x^4 \, and \, x < 1 \Rightarrow - x^9 + x^4 > 0 \, and \, 1 - x > 0$
$\therefore$ $x^{12} - x^9 + x^4 - x + 1 > 0, \, \forall \, 0 < x \le 1$ \hspace21mm ...(ii)

Case III When x > 1 $\Rightarrow x^{12} > x^9 \, and \, x^4 > x$
$\therefore x^{12} - x^9 + x^4 - x + 1 > 0, \forall \, x > 1$ \hspace21mm ...(iii)
From Eqs. (i), (ii) and (iii), the above equation holds for all x $\in$ R
Hence , option (d) is the correct answer.