Local Extrema of Real-Valued Functions
1. Single Variable Case
* Basic terminologies: local minimum, local maximum, absolute minimum, absolute
maximum, critical point, reflection point;
* First-order derivative test: x0 is a local extremum ⇒ f 0 (x0 ) = 0;
* Second-order derivative test: if f 0 (x0 ) = 0, then x0 is a local minimum if f 00 (x0 ) >
0; is a local maximum if f 00 (x0 ) < 0; inconclusive if f 00 (x0 ) = 0;
2. Multi-variable Case
• Some theory from linear algebra:
* Def (Quadratic function): g : Rn → R is a quadratic function if it has the
form
n
X
g(h1 , h2 , ..., hn ) =
aij hi hj = hAhT ,
i,j=1
where h = [h1 , h2 , ..., hn ] and A = (aij ) is an n × n matrix.
* Def (Positive-definite matrix): A symmetric n × n matrix A = (aij ) is positive definite if the quadratic function g associated with A is positive for every
non-zero h.
* Def (Negative-definite matix): A symmetric n × n matrix A = (aij ) is negative definite if the quadratic function g associated with A is negative for every
non-zero h.
* Fact 1: Let B =
a b
b c
, then B is positive definite if a > 0 and det(B) =
ac − b2 > 0.
* Def (Hessian matrix of a function): Suppose that f : U ⊂ Rn → R has
second-order continuous derivatives in U . Then, the Hessian matrix of f is
defined by


∂2f
∂2f
...
∂x1 ∂xn

 ∂x1 ∂x1
Hf =  ...
.
2
2
∂ f
∂ f
... ∂xn ∂xn
∂xn ∂x1
Note: Hf is a symmetric n × n matrix.
• Extrema of real-valued functions with multi-variables:
* Def (local minimum): Given f : U ⊂ Rn → R, a point x0 ∈ U is called a
local minimum of f is there is a neighborhood V of x0 such that for all points
x in V , f (x) ≥ f (x0 ).
Similarly, we can define local maximum;
* Def (critical point): A point x0 is a critical point of f if either f is not differentiable at x0 , or if it is, Df (x0 ) = 0.
* Def (saddle point): A critical point that is not a local extremum is called a
saddle point.
* Theorem 4 (section 3.3 First-derivative test for local extrema) If U ⊂ Rn is
open, the function f : U ⊂ Rn → R is differentiable, and x0 ∈ U is a local
extremum, then Df (x0 ) = 0; that is, x0 is a critical point of f .
Remark: Df (x0 ) = 0 is equivalent to ∇f = 0 i.e.
at x0 .
∂f
∂x1
=
∂f
∂x2
= .... =
∂f
∂xn
=0
* Theorem 5 (section 3.3 Second-derivative test for local extrema) If f : U ⊂
Rn → R is of class C 3 , x0 ∈ U is a critical point of f , and the Hessian Hf (x0 )
is positive-definite, then x0 is a relative minimum of f . Similarly, if Hf (x0 )
is negative-definite, then x0 is a relative maximum.
(Proof follows from Taylor’s formula)
* Theorem 6 (section 3.3 Second-derivative test for functions of two variables)
Let f (x, y) be of class C 2 on an open set U in R2 . A point (x0 , y0 ) is a (strict)
local minimum of f provided the following three conditions hold:
(1)
(2)
(3)
∂f
(x0 , y0 ) = ∂f
(x0 , y0 ) = 0;
∂x
∂y
2
∂ f
(x0 , y0 ) > 0;
∂x2
2
2 2 ∂f
D = ∂∂xf2 ∂∂yf2 − ∂x∂y
>
0 at (x0 , y0 );
Remark: (1) if (x0 , y0 ) is a local maximum if we change the sign in (2) to
negative;
(2) if D ≤ 0 at (x0 , y0 ), then we say (x0 , y0 ) is a saddle point;
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