# Extrema of a Function of Two or More Variables (without Constraint)

• Delia Koo
Part of the Heidelberg Science Library book series (HSL)

## Abstract

Let f(x 1, x 2) be a function of two real variables x 1 and x 2 with continuous partial derivatives
$${f_1} = \frac{{\partial f\left( {{x_1},{x_2}} \right)}}{{\partial {x_1}}}\,and\,{f_2} = \frac{{\partial ({x_1},{x_2})}}{{\partial {x_2}}}$$
(2.1.1)
Suppose that f(x 1, x 2) attains a local extremum at the point (a,b). Then intuitively it is clear that the function of a single variable f(x 1,b) must attain an extremum at x 1 = a. From Section 1.4 it is necessary that f 1 = 0 at x 1 = a. Similarly, since the function f(a,x 2) must also attain an extremum at x 2 = b, it is also necessary that f 2 = 0 at x 2 = b.

## Keywords

Partial Derivative Quadratic Form Maximum Likelihood Estimator High Derivative Local Extremum
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.