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# USE OF DIFFERENTIALS OF VARIOUS ORDERS IN THE STUDY OF MAXIMA AND MINIMA OF FUNCTIONS OF SEVERAL VARIABLES.

• Dennis M. Cates
Chapter

## Abstract

Let $$u=f(x, y, z, \dots )$$ be a function of the independent variables $$x, y, z, \dots ,$$ and set, as in the tenth lecture,
\begin{aligned} f(x+\alpha dx, y+\alpha dy, z+\alpha dz, \dots )=F(\alpha ). \end{aligned}
So that the value of u relative to certain particular values of $$x, y, z, \dots$$ is either a maximum or a minimum, it will be necessary and sufficient that the corresponding value of $$F(\alpha )$$ always becomes a maximum or a minimum, by virtue of the assumption $$\alpha =0. \$$ We conclude (see the tenth lecture) that the systems of values of $$x, y, z, \dots ,$$ which, without rendering discontinuous one of the two functions, u and du,  generates for the first, a maxima or a minima, and necessarily satisfies, regardless of $$dx, dy, dz, \dots ,$$ the equation
\begin{aligned} du=0, \end{aligned}
.

## Copyright information

© Springer Nature Switzerland AG 2019

## Authors and Affiliations

1. 1.Sun CityUSA