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Set $S := \{x \in \Bbb R\,:\, x < 3/x\}$.

(a) Determine whether $\sup S$ exists, and determine its value if it exists. Justify your answer.

(b) Determine whether $\inf S$ exists, and determine its value if it exists. Justify your answer.

I have worked out graphically that the solution of the set is $(-\infty,-\sqrt3)\cup(0,\sqrt3)$, but graphical solutions aren't allowed for this question.

Am I right to say that $\sup S = \sqrt3$? I have proven that $\sup S$ exists, how do I show that it is indeed $\sqrt3$?

How should I show that $S$ is not bounded below?

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  • $\begingroup$ You may have a "chicken or the egg" problem if you call that number $\sqrt{3}$. What is $\sqrt{3}$? $\endgroup$ – parsiad Feb 12 '16 at 14:45
  • $\begingroup$ Your solution set is correct, but how did you find this? Only in a graphical way? If you can (algebraically) solve $x<3/x$, you'll find the same solution set and you don't need a graphical approach. $\endgroup$ – StackTD Feb 12 '16 at 14:47
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Hint: Multiply both sides by $x^2$ to get $x^3\lt 3x$, then subtract $3x$ from both sides to get $x^3-3x\lt0$, and factor to get $x(x^2-3)\lt0$.

From here you need to eliminate $x=0$ as a solution, and then consider $x$ as either positive or negative, and greater or less than $\sqrt 3$.

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