how can i prove Bernoulli's inequality?

Question

Could you help me prove Bernoulli's inequality:

For all $x\geq -1$ and integers $r\geq 0$, $(1+x)^r\geq 1+rx$

using the relationship between arithmetic and geometric means?

Answer 1

Try setting up the weighted arithmetic-geometric mean inequality with the right variables, then manipulating it until it is of a form similar to that of the Bernoulli inequality. Then you're a substitution away.

Start with $\frac{\lambda_1 a + \lambda_2 b} {\lambda_1 + \lambda_2} <= $ ...

and let $a=1$ and $b=1+x$. If you get stuck, the Wikipedia article on Bernoulli's inequality has what you want.

Answer 2

Apply the AM-GM inequality to $r$ terms: $1$ taken $r-1$ times and $1+rx$ taken once. Start from the geometric mean.