How can I find the minimum of $x^2+\frac{a}{x}$ on $\mathbb{R}_+$ without calculus?
Minimum of $x^2+\frac{a}{x}$ without Calculus
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analysis
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2Why do you want to avoid calculus and make it yourself hard? – 2011-02-06
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0I think you want $a\geq0$. – 2011-02-06
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5@Jonas: the solution without calculus is actually much easier, as bobobinks shows. The use of AM-GM is a justification of a typical intuition in these situations: if you want to optimize f(x) + g(x) where f is increasing and g is decreasing, you roughly want a value of x where f and g are approximately equal. I find this much more enriching than a computation of derivatives. In general it is an interesting exercise to split up a sum in such a way that 1) AM-GM applies and 2) the equality case actually occurs. – 2011-02-06
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0If a<0 you have a problem: 1/x will get near -infinity when x tends towards 0. – 2011-02-06
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0@Qiaochu: You're right. Thanks for the advice. – 2011-02-06
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0@Jonas: Your comment is interesting since I go the other way around. Usually when I want to find a maximum or minimum the first thing I go after is some sort of bound obtained from simple inequalities. If I don't get it, my final resort is to use the big hammer, Calculus. – 2011-02-06
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0@Sivaram: I will try that too from now on, the reply is some kind of eye-opener. I would use calculus because it one of the first things I have learnt I think and it would be a no-brainer. – 2011-02-06
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0@Jonas: Thanks for taking my previous comment in the right spirit. The reason for my previous comment is that you do not need a hammer to kill a mosquito. And in general there is an elegance to solve problems without advanced tools. – 2011-02-06