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How to use AM-GM to find the minimum of a function

  1. Mar 22, 2017 #1
    1. The problem statement, all variables and given/known data
    Let ##\displaystyle f(x) = 3 \left( x + \frac{1}{x} \right)##. Use AM-GM to find the minimum value.

    2. Relevant equations


    3. The attempt at a solution
    I know how to find the minimum value using calculus: we take the derivative, set it to zero,then find the critical points. Then we can use the second derivative to distinguish between maxima and minima.

    However, I am not sure how to use the AM-GM inequality to do this, which is that ##\displaystyle \frac{x+y}{2} \ge \sqrt{xy}## for all non-negative real numbers x and y
     
  2. jcsd
  3. Mar 22, 2017 #2

    haruspex

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    Your f(x) includes the sum of two elements; the AM-GM formula includes a sum. Try something really obvious on that basis.
     
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