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  4. The largest interval for which x12 - x9 + x4 - x + 1 > 0 is

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

IIT JEEIIT JEE 1982Complex Numbers and Quadratic Equations

Solution:

$ x^{12} - x^9 + x^4 - x + 1 > 0 $
Here, three cases arises:
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 $ ...(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 $ ...(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 $ ...(iii)
From Eqs. (i), (ii) and (iii), the above equation holds for all $x \in$ R
Hence , option (d) is the correct answer.