ABSTRACT
For f involving several variables, the condition for f to have a relative minimum is more complicated. First, Equation (9.1)
∂ ∂
= ∂ ∂
= = ∂ ∂
=f x
P f x
P f x
( ) ( ) ( )� 0
(9.1)
must be satisfied. Second, the quadratic form (Equation 9.2)
Q f
x x P x x P x x P
j j= ∂
∂ ∂ - -
( )( ( ))( ( ))
(9.2)
must be positive for all choices of xi and xj in the vicinity of point P, and Q = 0 only when xi = xi (P) for i = 1, 2, …, n. This condition comes from a Taylor Series expan-
sion of f(x1, x2, …, xn) about point P using only terms up to ∂
∂ ∂ 2 f
x x P
( ). This gives
f x x x f P f x
P x x P
f
i i( , ,..., ) ( ) ( )( ( ))1 2
= + ∂ ∂
-
+ ∂ ∂
x x P x x P x x P
j j∂ - -
( )( ( ))( ( ))
If f (x1, x2,…, n) has a relative minimum at point P, then ∂ ∂
=f x
P i
( ) 0 for i = 1, 2,…, n and f (x1, x2,…, xn) – f (P) > 0 for all (x1, x2,…, xn) in the vicinity of point P. But f (x1, x2,…, xn) – f (P) = Q. Thus, for f (x1, x2, …, xn) to have a relative minimum at point P, Q must be positive for all choices of xi and xj in the vicinity of point P.
