\input zb-basic
\input zb-matheduc
\iteman{ZMATH 06672978}
\itemau{Varga, Marek; Michali\v{c}ka, Peter}
\itemti{Arithmetic mean and geometric mean.}
\itemso{Acta Math. Nitriensia 2, No. 2, 43-48 (2016).}
\itemab
Summary: In mathematics we define several types of means. Probably the most known are the arithmetic and geometric means. If we are given $n$ nonnegative numbers $x_1$, $x_2ยง, \dots, $x_n$; then the number $A_n=\frac{x_1+x_2+\dots +x_n}{n}$ we call the arithmetic mean and the number $G_n=\sqrt[n]{x_1x_2\cdots x_n}$ we call the geometric mean of the numbers given. In the first part of the paper with the use of functions' their of more variables we will show that for each natural number $n$ $A_n\ge G_n$ applies. In the second part we will try to show the same, however, without using the differential calculus.
\itemrv{~}
\itemcc{I20 I40 F60}
\itemut{arithmetic mean; geometric mean; proof; differential calculus}
\itemli{doi:10.17846/AMN.2016.2.2.43-48}
\end