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

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


Let f(x 1, x 2) be a function of two real variables x 1 and x 2 with continuous partial derivatives
$$ {f_1} = \frac{{\partial f\left( {{x_1},{x_2}} \right)}}{{\partial {x_1}}}\,and\,{f_2} = \frac{{\partial ({x_1},{x_2})}}{{\partial {x_2}}} $$
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.


Partial Derivative Quadratic Form Maximum Likelihood Estimator High Derivative Local Extremum 
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Copyright information

© Springer Science+Business Media New York 1977

Authors and Affiliations

  • Delia Koo
    • 1
  1. 1.Eastern Michigan UniversityYpsilantiUSA

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