If $x^2+\frac{1}{x}+6<0$ where $x\in \mathbb{R}$ then find the maximum value of $x$.
(I tried using AM-GM inequality, but couldn't establish a relation that would lead to the maximum value of $x$)
finding x based on inequality
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inequality
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0the second one. – 2017-01-16
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0I guess, you meant, find the maximum value of expression (which in this case is unbounded). Also, just to add you cannot directly apply AM-GM inequality because $x \in \mathbb{R}$ and AM-GM is applicable for non-negative quantities only. – 2017-01-16
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0nah, max value of x – 2017-01-16
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1There is no max value of $x$ since the inequality is obviously satisifed when $x \to \infty\,$. – 2017-01-16
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0In that case, the given expression is satisfied for all positive $x$ obviously. Sure that you aren't missing out some other constraints? – 2017-01-16
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0I've edited the question take a look now – 2017-01-16
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0There is no maximum value of $x$ for the reverse inequality as edited, either. $\sup \{x \in \mathbb{R} \mid x^2 + \frac{1}{x}+6 \lt 0 \} = 0$ but the maximum $x=0$ is not attained since the inequality is not defined for $x=0$. Was the question the maximum ***absolute*** value of $x$ maybe? – 2017-01-16