Week 4 exercises, due 19 October 2021

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Exercises 4.6 Week 4 exercises, due 19 October 2021

1.

Suppose \(x,y>0\) and \(r\) is a rational number so that \(0\lt r\leq 1\text{.}\) Use the AM/GM inequality to prove

\begin{equation*} x^{r}y^{1-r} \leq rx+(1-r)y\text{.} \end{equation*}

2.

Prove that if \(m_{1},...,m_{k}\) are positive integers and \(y_{1},...,y_{k}>0\text{,}\) then

\begin{equation*} \frac{m_1y_1+m_2y_2+\cdots+m_ky_k}{m_1+m_2+\cdots+m_k} \geq \sqrt[ m_1+m_2+\cdots+m_k]{y_1^{m_1}\cdot y_2^{m_2}\cdots y_k^{m_k}} \end{equation*}