Question

The largest interval among the following 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 $-\infty <x<\infty$

$\Rightarrow$ If $x\le 0$, then $x^{12}-x^9+x^4-x+1>0$

$\Rightarrow$ If $0<x\le 1$, then $x^{12}+x^4(1-x^5)+(1-x)>0$

$\Rightarrow$ If $x>1$, then $x^9(x^3-1)+x(x^3-1)+1>0$

So the expression $x^{12}-x^9+x^4-x+1>0$ is valid for $-\infty <x<\infty$