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

The Correct Option is D

$ 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.