8
$\begingroup$

How can I find the minimum of $x^2+\frac{a}{x}$ on $\mathbb{R}_+$ without calculus?

  • 2
    Why do you want to avoid calculus and make it yourself hard?2011-02-06
  • 0
    I think you want $a\geq0$.2011-02-06
  • 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
  • 0
    If a<0 you have a problem: 1/x will get near -infinity when x tends towards 0.2011-02-06
  • 0
    @Qiaochu: You're right. Thanks for the advice.2011-02-06
  • 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
  • 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
  • 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

5 Answers 5