How do I find range using AM-GM inequality

Question

I have been given a function, $$ f(x)= x^2+ \dfrac{9}{x^2} $$ and I been told to find the range of this function using AM-GM inequality only.

I was able to calculate minimum value of $f(x)$ which is $6$

$$ x^2 + \dfrac{9}{x^2} ≥ 6$$

and by setting $x^2 = \dfrac{9}{x^2}$ I get min value as $6$

But how do I calculate maximum value for the range?

Answer 1

Because of $x^2$ and the positivity of $\frac9{x^2}$, the function has no maximum value (it is unbounded above as $x\to\infty$ or $x\to0$).

Answer 2

No maximum exists there since $$f(x)>x^2$$which means that $f(x)$ can be arbitrarily large. Here is a sketch