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

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.