Elements of Optimization pp 13-28 | Cite as

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

Chapter

## Abstract

Let Suppose that

*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)

*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.

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## Copyright information

© Springer Science+Business Media New York 1977