Problem 171 (difficulty: 6/10)

Prove that \(\displaystyle (1+x)^{r}\leq 1+rx\) if \(\displaystyle r\in\Q\), \(\displaystyle 0<r<1\) and \(\displaystyle x\geq -1\).

Solution:

\(\displaystyle r=p/q,\ \sqrt[q]{(1+x)^p\cdot 1^{q-p}}{\leq}\frac{p(1+x)+(q-p)}q\).

Give me another random problem!