Taylore Series of several variables

Question 1

Taylor series for a function of one variable

Answer 1

f(x) = f(a) + ∑(from k=1 to inf) f^(k)(a)/k! (x-a)^k
It can also be written as
f(x) = f(a) + ∑(from k=1 to inf) 1/k! (u ∂/∂x)^k f(a)

Question 2

Taylor series for two variables

Answer 2

The taylor series for a two variable function f(x,y)=f(au, b+v), u=x-a, v=y-b is

f(a+u, b+v) = f(a,b) + ∑(from k=1, to inf) 1/k! (u ∂/∂x + v ∂/∂y) ^k f(a,b)

Question 3

Taylor series for a three variables

Answer 3

The taylor series for a three variable function f(x,y,z) = f(a+u, b+v, c+w) = f(a,b,c) + ∑(k=1 to inf) 1/k! [ u ∂/∂x + v ∂/∂y + w ∂/∂z)^k f(a,b,c)

Question 4

Tips for Taylor Series

Answer 4

Use known expansions where ever possible

Question 5

Local maximum (of a one variable function)

Answer 5

Let f: D C |R^n -> |R. The point a e D is said to be a local maximum if f(x) =< f(c) for all x sufficiently close to a

Question 6

Local minimum (of a one variable function)

Answer 6

Let f: D C |R^n -> |R. The point a e D is said to be a local minimum if f(x)>=f(c) for all x e D sufficiently close to a

Question 7

Global maximum (of a one variable function)

Answer 7

Let f: D C |R^n -> |R. The point a e D is said to be a local maximum if f(x) =< f(c) for all x e D

Question 8

Global minimum (of a one variable function

Answer 8

Let f: D C |R^n -> |R. The point a e D is said to be a local minimum if f(x)>=f(c) for all x e D

Question 9

Local extremum

Answer 9

If it is a local minimum or maximum

Question 10

Global extremum

Answer 10

If it a global minimum or maximum

Question 11

Stationary or critical point

Answer 11

Let f: D C |R^n -> |R. The point a e D is a stationary or critical point if ▽f(a) =0

Question 12

A singular point

Answer 12

Let f: D C |R^n -> |R. The point a e D is a singular point if ▽f does not exist at a.

Question 13

Necessary conditions for extremum

Answer 13

Let D C |R^n -> |R. If f has a local extremum at the point a e D, then a must be either

(1) a stationary point of f or
(2) a singular point of f or
(3) a boundary point of D

Question 14

Saddle point

Answer 14

A critical point a which is neither a local maximum nor a local minimum is called a saddle point

Question 15

Hessian matrix (of 2 variables)

Answer 15

If f is a real function of two variables, and all the second order partial derivatives of f exist at the point (a,b) then the Hessian matrix of f at (a,b) is defined as
H= (fxx fxy)
(fyx fyy)

Question 16

Hessian matrix (of 3 variables)

Answer 16

Hessian matrix of f at the point (a,b,c) is
H = (fxx fxy fxz)
(fyx fyy fyz)
(fzx fzy fzz)

Question 17

The leading minor test (two variables)

Answer 17

Suppose f is a sufficiently smooth function of two variables with a critical point at (a,b) and H the Hessian matrix of f at (a,b) is denoted as follows
H=(fxx fxy) det(H) = fxx fyy - (fxy)^2
(fxy fyy)
where the partial derivatives in H are evaluated at (a,b)

Question 18

The leading minor test (outcomes) (two variables)

Answer 18

If det(H) > 0 and fxx < 0, then (a,b) is a local maximum
If det(H) > 0 and fxx > 0, then (a,b) is a local minimum
If det(H) < 0, then (a,b) is a saddle point
If det(H) =0, then we need to investigate further.

Question 19

Leading minors of H

Answer 19

Let H be an nxn -matrix and for each 1≤ m≤n, let Hm be the mxm-matrix formed from the first m rows and m columns of H. The determinants det(Hm), 1≤m≤n are called the leading minors of H.

Question 20

The leading minor test

Answer 20

If f: |R^n -> |R, then a stationary point x0 where det(H) ≠ 0 is

(1) a local maximum if (-1)^m det(Hm) >0 for all m, 1≤m≤n
(2) a local minimum if det(Hm) >0 for all m , 1≤m≤n
(3) a saddle point if none of the above.

If you cannot apply the test, you must classify a stationary point directly from the definition.

Question 21

Boundedness Theorem

Answer 21

If f is a continuous function of n variables on a closed bounded domain of |R^n then f is bounded and attains its global maximum and minimum

Question 22

Constrained extreme value problem

Answer 22

Maximise/minimise f(x,y)

Subject to the constraint g(x,y) = 0

Question 23

Lagrangian function (Theory)

Answer 23

We can look for critical points of the Lagrangian function to find points on the curve g(x,y) =0 such that f(x,y) is maximum or minimum.

Question 24

Lagrangian function

Answer 24

L(x, y, λ) = f(x,y) + λg(x,y)

Question 25

Stationary equations of the Lagrangian function

Answer 25

At any critical point of L we must have
Lx = fx + λgx=0
Ly = fy + λgy=0
Lλ = g(x,y) = 0

This says that ▽f is parallel to ▽g.