Bernoull’s Inequality

Theorem (Bernoulli's Inequality) Let \(\alpha\) be a positive real number and \(\delta\geq -1\). If \(0< \alpha\leq 1\), the \( (1+\delta)^\alpha \leq 1+\alpha\delta, \) and if \(\alpha\geq 1\), then \( (1+\delta)^\alpha\geq 1+\alpha\delta. \) Proof. Assume \(0< \alpha\leq 1\). Let \(f(x)=x^\alpha\). By the mean value theorem, \( f(1+\delta)=f(1)+\alpha\delta c^{\alpha-1} \) for some \(c\) between \(1\) and \(1+\delta\). If \(\delta>0\), then \(c>1\). Since…