Maxima and Minima

• Phil Dyke
Chapter
Part of the Macmillan College Work Out Series book series (CWOS)

Abstract

A function of two variables f(x, y) when written in the form z = f(x, y) represents a surface in three-dimensional space; it is this that provides the principal application of the theory that follows. Taylor’s Theorem in two variables takes the form:
$$f(a + h,{\mkern 1mu} b + h) = f(a,{\mkern 1mu} b) + \left( {h\frac{\partial }{{\partial x}} + k\frac{\partial }{{\partial y}}} \right)f(a,{\mkern 1mu} b) + \frac{1}{{2!}}{\left( {h\frac{\partial }{{\partial x}} + k\frac{\partial }{{\partial y}}} \right)^2}f(a,{\mkern 1mu} b) + \ldots {\text{ }}\frac{1}{{n!}}{\left( {h\frac{\partial }{{\partial x}} + k\frac{\partial }{{\partial y}}} \right)^n}f(a,{\mkern 1mu} b) + {R_n}$$
where $$\frac{1}{{r!}}{\left( {h\frac{\partial }{{\partial x}} + k\frac{\partial }{{\partial y}}} \right)^r}f(a,{\mkern 1mu} b)$$ is interpreted as the operator $$\frac{1}{{r!}}{\left( {h\frac{\partial }{{\partial x}} + k\frac{\partial }{{\partial y}}} \right)^r}$$ acting on the function f(x, y) then x placed equal to a and y placed equal to b (r = 1, 2, …, n). R n is the remainder term.