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

1. Mar 22, 2017

### Mr Davis 97

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. Mar 22, 2017

### haruspex

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