Theory of Equations 5 Question 7
7. The largest interval for which $x^{12}-x^{9}+x^{4}-x+1>0$ is
$(1982,2 M)$
(a) $-4<x \leq 0$
(b) $0<x<1$
(c) $-100<x<100$
(d) $-\infty<x<\infty$
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Answer:
Correct Answer: 7. (d)
Solution:
- Given, $x^{12}-x^{9}+x^{4}-x+1>0$
Here, three cases arises:
Case I When $x \leq 0 \Rightarrow x^{12}>0,-x^{9}>0, x^{4}>0,-x>0$
$\therefore \quad x^{12}-x^{9}+x^{4}-x+1>0, \forall x \leq 0$
Case II When $0<x \leq 1$
$ \begin{aligned} & x^{9}<x^{4} \text { and } x<1 \Rightarrow-x^{9}+x^{4}>0 \text { and } 1-x>0 \\ & \therefore \quad x^{12}-x^{9}+x^{4}-x+1>0, \forall 0<x \leq 1 \end{aligned} $
Case III When $x>1 \Rightarrow x^{12}>x^{9}$ and $x^{4}>x$
$\therefore \quad x^{12}-x^{9}+x^{4}-x+1>0, \forall x>1$
From Eqs. (i), (ii) and (iii), the above equation holds for all $x \in R$.
