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

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© Springer Science+Business Media New York 1977