Question

How do you use the Mean Value Theorem to prove Bernoulli's inequality?

Answer 1

The Mean Value Theorem for `f(x)=(1+x)^n` on `[0,x]` gives that

`[(1+x)^n-1]/x=n(1+c)^{n-1}`

Because `n*(1+c)^(n-1)>n` hence

`[(1+x)^n-1]/x=n(1+c)^{n-1}>n=>(1+x)^n-1>n*x=>(1+x)^n>1+nx`

which completes the proof.

The case for `−1≤x<0` is similar. The case for `x=0` is simple.