2
1 Preliminaries
expansion. Equation (1.1.2) is called the Lagrange’s remainder. When x → x 0 ,
R n is a higher order infinitesimal than |x − x 0 |
n , or R n is a higher infinitesimal of
order n − 1 than |x − x 0 |.
Theorem 1.1.2 Suppose that a derivable function f (x) has an extremal value at a
point x 0 in the interval of definition, then there must be f
(x 0 ) = 0. The theorem is
called the extremal theorem of function of one variable.
1.1.2 Cases of Functions of Several Variables
The Taylor mean value theorem of a function of one variable can be extended to the
case of function of several variables. Below the Taylor formula of binary function
which has Lagrange’s remainder will be discussed first.
Theorem 1.1.3 Suppose that a function f (x, y) is continuous in a convex neighborhood at point (x 0 , y 0 ) and has until the continuous partial derivative of order n
+1, and let x = x 0 + x, y = y 0 + y be an arbitrary point within the neighborhood, then there is always a θ (0 < θ < 1), to make the following Taylor formula of
order n
f (x, y) = f (x 0 , y 0 ) +
x
∂
∂ x
+ y
∂
∂ y
f (x 0 , y 0 ) +
1
2!
x
∂
∂ x
+ y
∂
∂ y
2
f (x 0 , y 0 )
+ · · · +
1
k!
x
∂
∂ x
+ y
∂
∂ y
k
f (x 0 , y 0 ) + · · · +
1
n!
x
∂
∂ x
+ y
∂
∂ y
n
f (x 0 , y 0 ) + R n
(1.1.3)
hold, where, the general term is
x
∂
∂ x
+ y
∂
∂ y
k
f (x 0 , y 0 ) =
k
r =0
C
r
k ((x)
r
((y)
k−r ∂
k f (x 0 , y 0 )
∂ x r ∂ y k−r
(1.1.4)
namely according to Newton binomial theorem, to expand it into the summer of
k + 1 terms, here, C
r
k =
k!
r !(k−r )!
is a combination number to take r elements from k
elements. The remainder is
R n =
1
(n + 1)!
x
∂
∂ x
+ y
∂
∂ y
n+1
f (x 0 + θθx, y 0 + θθy)
(1.1.5)
where, R n is called the nth order Lagrange(’s) remainder of f (x, y) at point
(x 0 , y 0 ).
Theorem 1.1.3 is called the Taylor mean value theorem of function of two
variables.