<s>[INST] How do I solve the inequality $x^{12}−x^9+x^4−x+1>0$ using intervals? My book has the question $x^{12}−x^{9}+x^{4}−x+1>0$. The solution given gives three cases, when $x \le 0$, when $0 < x \le 1$ and when $x > 1$. How did they get these intervals? What is the method used?
 [/INST] Clearly for $x=0$, We have $$x^{12}-x^{9}+x^4-x+1>0$$
Now for $x\neq 0$, Using Arithmetic Geometric Inequality
$$\frac{x^{12}+x^{12}+x^{12}+1}{4}\geq \sqrt[4]{x^{12}\cdot x^{12}\cdot x^{12}\cdot 1}=|x^9|>x^9$$
$$\frac{x^4+1+1+1}{4}\geq \sqrt[4]{x^4\cdot 1 \cdot 1\cdot 1}=|x|>x$$
So $\displaystyle \frac{3x^{12}}{4}-x^9+\frac{x^4}{4}-x+\frac{1}{4}+\frac{3}{4}>0$(Equality hold when $x=1$)
So $$ x^{12}-x^9+x^4-x+1>\frac{3x^{12}}{4}-x^9+\frac{x^4}{4}-x+1>0\forall  x\in\mathbb{R}.$$
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