<s>[INST] How to find the Max, Min,Sup and inf of $A = \left \{ \frac{1}{3^\frac{x}{2}+3^\frac{2}{x}} \right \}$ I need to find the min, max, sup and inf of $$A = \left \{ \frac{1}{3^\frac{x}{2}+3^\frac{2}{x}} , x>0, x \in \mathbb{R} \right \}$$
My attempt:
$3^\frac{x}{2} + 3^\frac{2}{x}  \geqslant 2\sqrt{3^{2 \left | x \right |}} = 2 \times {3^x}$ By the A.M G.M inequality. Also:
$$ 0 \leqslant \frac{1}{3^\frac{x}{2}+3^\frac{2}{x}} \leq \frac{1}{2 \times 3^x} $$
I'm a little stuck now and I would appreciate some assistance.
 [/INST] Consider the function $f(x)=\frac{1}{3^\frac{x}{2}+3^\frac{2}{x}}$ defined on $(0,\infty).$ 


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*Clearly, $f(x)>0$ for any $x>0.$ Since $\lim_{x \to 0^+}f(x)=0\quad$ and $\quad\lim_{x \to \infty}f(x)=0,$ $$Inf A=0 \quad \text{and A doesn't have a minimum}.$$

*If $x \in (0,2),$ then $3^\frac{x}{2}<3^\frac{2}{x}$ and $$f(x)<\frac{1}{2 \cdot 3^{x\over 2}}<\frac{1}{6}$$
If $x \in (2,\infty),$ then $3^\frac{2}{x}<3^\frac{x}{2}$ and $$f(x)<\frac{1}{2 \cdot 3^{2\over x}}<\frac{1}{6}$$
Thus $1\over 6$ is un upper-bound of $A.$ Since $f(2)={1\over 6},$ this is the maximum of $A.$
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