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Lesson 7: Differentiability and Optimization, Apuntes de Administración de Empresas

Universidad Rey Juan Carlos (URJC)Administración de Empresas
Prof. Clement Kanyida-Malu Kabiena

Asignatura: corporate mathematics, Profesor: Clement Kanyida-Malu Kabiena, Carrera: Business Administration and Management, Universidad: URJC

Tipo: Apuntes

2017/2018

Subido el 09/01/2018

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LESSON 5. DIFFERENTIABILITY
Mathematics – 2011/2012 - 1 -
LESSON 6
DIFFERENTIABILITY
1. Introduction
Let
,:
ℜ
→
ℜ
f
be a function. If the first h derivatives
)(),...,(
00
xfxf
h
′
exist then we have
got some information about the function’s behaviour in the neighbourhood of x.
How to extend this consideration to
mn
fℜ→ℜ;
?
We already know that the gradient
∂
∂
∂
∂
=∇ )(),...,()(
00
1
0
x
x
f
x
x
f
xf
n
corresponds to the first
derivative in multiple variables functions, so it is quite natural that the gradient must
provide the same information as
),(xf
′
about the local behaviour of the function.
If f , function defined in
ℜ
, is derivable at
fx ⇒
0
is continuous at
0
x
. However, in
2
ℜ
the function
2
2 4
( , ) (0,0)
( , )
0 ( , ) (0,0)
xy x y
f x y x y
x y
≠
=+
=
is derivable in all directions, but it is not
continuous at
)0,0(
.
So, we need to extend the derivability when working in more than one dimension.
Definition of differential
By definition
ℜ
→
ℜ
:f
is derivable at a if there exists
)(
)()(
lim
0
af
h
afhaf
h
′
=
−
+
→
Or, in the same way
0
)()()(
0
→
⋅
′
−
−
+
→h
h
hafafhaf
We call
hafhl
⋅
′
=
)()(
Remark
1. If
)(af
′
is a constant
)(hl
is a linear function, where
0)0(
=
l
.
2.
)()()( hlafhaf
≈
−
+
3. l can be interpreted as a function which approximates linearly the increase of f
in a neighbourhood of a.
pf3
pf4
pf5

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LESSON 6

DIFFERENTIABILITY

1. Introduction

Let f : ℜ→ℜ,be a function. If the first h derivatives f ( x 0 ),..., f ( x 0 )

′ h

exist then we have

got some information about the function’s behaviour in the neighbourhood of x.

How to extend this consideration to

n m

f ; ℜ →ℜ?

We already know that the gradient 

1

0 x x

f x x

f f x n

corresponds to the first

derivative in multiple variables functions, so it is quite natural that the gradient must

provide the same information as f ′(^ x ),about the local behaviour of the function.

If f , function defined in ℜ , is derivable at x 0 ⇒ f is continuous at x 0. However, in

2 ℜ

the function

2

2 4 ( ,^ )^ (0, 0) ( , )

0 ( , ) (0, 0)

xy x y f x y x y

x y

is derivable in all directions, but it is not

continuous at ( 0 , 0 ).

So, we need to extend the derivability when working in more than one dimension.

Definition of differential

By definition f :ℜ→ℜ is derivable at a if there exists ( )

lim 0

f a h

f a h f a

h

→

Or, in the same way 0

h → h

f a h f a f a h

We call l ( h )= f ′( a )⋅ h

Remark

1. If f ′(^ a )is a constant ⇒ l ( h )is a linear function, where l ( 0 )= 0.

2. f ( a + h )− f ( a )≈ l ( h )

3. l can be interpreted as a function which approximates linearly the increase of f

in a neighbourhood of a.

Definition

f :ℜ→ ℜ is said to be differentiable at a if there exists a linear map l :ℜ→ℜsuch that

f ( a + h )− f ( a )= l ( h )+ α( a + h )⋅ h

And α( a + h ) h → 0 → 0

α is a function of the error made when approximating the increase of f through a

linear function.

To extend the same observation to :

n

f ℜ → ℜ is easy.

Let v be any direction of a vector. We will say that the differential of f at the point a

in the direction of v exists if also exists a linear map ℜ →ℜ

n

l : such that

f ( a +λ v )− f ( a )= f (λ v )+α( a +λ v ) λ v

where α ( a + λ v ) → λ → 0 0

Definition

n

f : is differentiable at point a if exists a linear map l such that f is

differentiable in any direction v.

Properties

We know that if f is differentiable, then f ( a ) v ,

v

∃ ′ ∀ particularly all partial derivatives

exist and f a lv f a v

v

Example

Let f x y = x + y + xy

2

( , ) be a function

Work out the differential of f at the point ( 1 , 2 )in the following directions

w

v

Example

Let f ( x , y )= ( xy ,ln( x + y )) be a function. Try to get df ( 1 , 1 )

1 (^1 ,^1 ) (^1 ,^1 )

1 1 = ∇ = ∂

x f y

f y x

f

2

2 2 ∇ =

f y x y

f

x x y

f

x y

x

x y

y

Jf ( x , y ) 1 1 

Jf ( 1 , 1 )

1 2 1 2 2

1 1 2

2 2

v v v v v

v v v

df

Properties of differential

1. If f is differentiable at a ⇒ df ( a )is unique

2. If f is differentiable at a ⇒ f is continuous at a

3. If f is differentiable at a ⇒ f a df a v v

v

4. If f : ℜ→ℜ f is differentiable ⇔ f is derivable

Definition

Let :

n

f ℜ →ℜ be a continuous function at a

f belongs to

1

C class if

xi

f i n ∂

∀ = 1 , 2 ,..., ∃ and they are continuous.

f belongs to

2

C class if

xi xj

f i n ∂ ∂

2

1 , 2 ,..., and they are continuous.

f belongs to

k

C class if 1, 2,...,

k

i ik

f i n x x

∂ K ∂

and they are continuous.

2. Sufficient condition of differentiability

If f belongs to

1

C class then f is differentiable at a.

Theorem of Schwartz

If

2 2 2 2 , i j i j j i

f f f f C i j and x x x x x x

If ( ),

2

f ∈ C ⇒ Hf x the Hessian matrix is symmetric.

Property

If f belongs to

2

C class at a f ( a ), v ,

v

⇒ ∃ ′′ ∀ then it can verified that:

i j

n

i

n

j (^) i j

t v

a vv x x

f f ( c ) v Hf ( a ) v ( ) 1 1

2

′′ = ⋅ ⋅ = where Hf ( a )is a symmetric matrix.

How to study the behaviour of a function at a point in the direction of a vector.

If f v ′( )^ a > 0 and f v ′′( )^ a > 0 the function increases acceleratedly at that point

If f v ′( )^ a < 0 and f v ′′( )^ a > 0 the function decreases deceleratedly at that point

If f v ′( )^ a < 0 and f v ′′( )^ a < 0 the function decreases acceleratedly at that point

If f v ′( )^ a > 0 and f v ′′( )^ a < 0 the function increases deceleratedly at that point

OPTIMIZATION

3. Non-constrained Optimization

Definition

f ( x ) reaches a local maximum at a if ∀ x in the neighbourhood of a it is verified

[ ( ) ( )]

f a f x

f a f x

≤

If inequialities are strict, maxima or minima are strict as well.

Remark

If f ( x )reaches a maximum at a ⇒ f ( a )≥ f ( a + λ v )for λ = λ 0

If f ( x )reaches a minimum at a ⇒ f ( a )≤ f ( a + λ v )for λ = λ 0

Critical points 

3

Hf

x

1

2

∂

x

f 0 2 1

2

∂ ∂

x x

f 2 3 1

2 = − ∂ ∂

x x

f

1 2

2

∂ ∂

x x

f

2

2 2

f

x

3 2

2

∂ ∂

x x

f

1 3

2 = − ∂ ∂

x x

f 0 2 3

2

∂ ∂

x x

f 2 3 3

2 6 x x

f

∂

Hf

1

2

3

det( ) 2

det( ) 4

det( ) 8

H

H

H

Indefinite

( 0 , 0 , 0 ) is a saddle point of f

Hf

1

2

3

det( ) 2

det( ) 4

det( ) 8

H

H

H

Positive definite

is a local minimum of f