7) Stationary Points Of Functions

Question 1

Stationary points for functions of more than one variables

Answer 1

Involve directional derivatives as for all directions ^u_ …

(grad φ)_p ….

Question 2

Local extremums are defined for:

Answer 2

For a ball of radius ρ around x_0:
Function C_1 f:S_1 -> S_2 f: x -> f(x) where s_1 subset of R^ n and s_2 R

The for a given point x_0 in S consider set
D_ ρ(x_0) = { x in S | ||x-x_0|| < ρ} for
ρ in R
Where ||x-x_0|| is distance

Question 3

Local extremums if

Answer 3

For a stationary point?

f(x) has a local maximum at x_0 if
There exists D_ρ(x_0) such that f(x) less than or equal to f(x_0) for all x in D_ρ(x_0)

f(x) has a local minimum at x_0 if
There exists D_ρ(x_0) such that f(x) more than or equal to f(x_0) for all x in D_ρ(x_0)

Otherwise. Saddle point if stationary?

Question 4

Nature of stationary points

Answer 4

E.g. (0,0) for example is a saddle point for a map when
f(x,0) bigger than 0 for all x not equal to 0
f(0,y) bigger than 0 for all y not equal to 0

• to explore nature of stationary point (x_0, y_0) explore if
f(x_0+h, t_0 +k) is bigger than or less than f(x_0,y_0) for all (h,k)
SHOWING THAT ITS A OBSL MAX OR MIN THEREFORE IS A LOCAL MAX OR MIN

• by Taylor’s, hessian or quadratic form

(grad φ)_p

Question 5

Taylor theorem
f(x +h) = f(x) + f’(x)h + (1/2!) f’‘(x)h^2 +…
+ (1/(n-1)!) f^(n-1) (x) h^(n-1) + R_n

Where R_n = (1/(n!) f^(n) (x+theta*h) h^(n)
For theta between 0 and 1

Gives theorem for two variables

Answer 5

Theorem for Taylor of two variables

Let f:S_ -> S_2, f:(x,y) -> f(x,y) where S_1 subset of R^2 and S_2 subset of R

For a point P(a,b) in S such that all partial derivatives of f up to and including order n exist and are continuous run an open region subset of S_1
INVOLVES Partial derivatives evaluated at (a,b)
Consider (h,k) in R^2 st (a+h, b+k) in S
Then
f(a+h, b+k) =f(a,b) + f_x h + f_y k + (1/2!) [ f_xxh^2 + 2f_xyhk + f_yy*k^2] +…

Question 6

Classifying stationary points by Taylors thereom

To determine the local behaviour in the vicinity of the stationary point

Answer 6

f(x_0 +h, y_0 +k) - f(x_0, y_0)

FOR A STARIONARY POINT Taylor’s the first partial derivatives become 0 so Taylor’s rearranged

Second order terms + HOT

Giving quadratic form
Q(h,k) = Ah^2 + 2Bhk + Ck^2
Where second order partials
A= fxx(x_0,y_0), B= fxy(x_0,y_0), C= fyy (x_0,y_0)

If A > 0, AC > B^2 implies that Q(h,k) bigger than or equal to 0 for all (h,k) thus
P is a local minimum

If A < 0, AC > B^2 implies that Q(h,k) less than or equal to 0 for all (h,k) thus
P is a local maximum

If AC < B^2 then Q(h,k) can take. Either sign and is a saddle point

If A=0 interchange C
If A=C=0 second order partials equal and is a SADDLE POINT
If AC=B^2 O rA=B=C=0 :
No conclusion and must consider further terms in Taylor’s series

Question 7

Questions on classifying stationary

Answer 7

Answers