Parametric inequality

Question

$k$ is a non negative real parameter. I have to study for which real $x$, $x^4+4-k^2$ is greater than zero and for which is $\leq 0$.

My prof says about the solutions that in the first case I have the ``big'' $x$ and $k<2$; in the latter, I have the little ones and $k\geq2$.

My question is: shouldn't I have solutions only for $k>2$? And how can I obtain these solutions?

Answer 1

Let consider

$$f(x)=x^4+4$$

and then study the conditions for

$$f(x)=x^4+4\ge h= k^2$$

$$f(x)=x^4+4\le h= k^2$$

with $h=k^2 \ge 0$.

Notably since $f(x)\ge 4$

• for $0<h\le4\implies f(x)\ge h \quad \forall x$
• for $h>4\\\implies f(x)\ge h \quad \forall x\in (-\infty,-\sqrt[4] h]\cup [\sqrt[4]h,+\infty)\\\implies f(x)\le h \quad \forall x\in [-\sqrt[4] h,\sqrt[4] h]$

Answer 2

write your inequality in the form $$x^4\geq k^2-4$$ if $$k\le 2$$ then our inequality is fulffiled. For $$k\geq 2$$ we get $$|x|\geq \sqrt[4]{k^2-4}$$ can you finish?