Question

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

• A. $[0,\infty )$
• B. $(-\infty ,0]$
• C. $(-\infty ,\infty )$
• D. None of these

Answer 1

The correct option is C $(-\infty ,\infty )$

Let $f\left(x\right)=x^{12}-x^9+x^4-x+1=x^9(x^3-1)+x(x^3-1)+1$
$=(x^9+x)(x^3-1)+1>0\forall x\ge 1$ ...(1)
Again, $f\left(x\right)=x^{12}-x^9+x^4-x+1=x^4(x^8+1)-x(x^8+1)+1$

$=x(x^8+1)(x^3-1)+1>0\forall x\le 0$ ...(2)
Next, $f\left(x\right)=x^{12}-x^9+x^4-x+1=x^{12}+x^4(1-x^5)+(1-x)>0$ for $0<x<1$ ...(3)
Combining (1),(2) and (3) we get $f\left(x\right)>0$ for $x\in (-\infty ,\infty ).$