Question

ECON 1111A/B Mathematical Methods in Economics II 2nd term, 2018-2019 Assignment 6 Show your steps clearly Define the definiteness of the following A-[1 5 a. b. 1 4 6 d. D= -2 3 1 -2 1 2 E 2 -3 1 2. Is the function f(x,y) - 7x2 + 4xy + y2 positive definite, negative definite, positive semidefinite or negative semidefinite? Find the extreme values for the following functions and identify whether they are local maximum, local minimum, and saddle points a. f(x, y) = x2 + xy + 2y2 + 3 b. f(x, y)--x2- y2 + 6x + 2y f(x, y) = x3-y3 4.Find the critical points, if any, of the following functions. Check whether they are relative maxima or relative minima or neither by using Hessian Matrix a. fx, y)--2x+x2-4y + xy y2 f(x, y, z)-x2 + y2 + 7z?-ху-3yz

Answer 1

1) a) Given

6 3

Now,

= [1], det(A1)-1 > 0

1 5-A, det(A) 6+15 21 0 213 6

Since these are all positive, the quadratic form induced by A is positive definite.

b) Given

B [3]det(B1)-0

A matrix B is negative definite if for odd and
det(B;) > 0
for even.

Since <0 and >0, we have B is a negative definite matrix.

c)Given

Now,

Neither of the condition of positive nor negative definite holds and determinant not equal to zero, C is indefinite.

d) Given

Now,

Hence D is positive definite.

e) Given

Now,

E,-[-2], det(E1)--2 < 0

E is Indefinite.

2. Given

Quadratic form is

thus the form is positive definite.

3. a)

Set first derivative equal to zero to get the critical points

thus, x=0, y=0. (0, 0) will be our critical point.

and

Therefore is a local minimum.

b)

Set first derivative equal to zero to get the critical points

thus, x=3, y=1. (3, 1) will be our critical point.

D(3,1)--2.-2-0 = 4 > 0
and

Therefore is local maximum.

c)

Set first derivative equal to zero to get the critical points

will be our critical point.

. Then is not a local maximum nor a local minimum

d)

Set first derivative equal to zero to get the critical points

(0, 0) will be our critical point

fxx =ー , fyyー2.fxyー0 2

f(0, 0)=0 is not a local maximum nor a local minimum.

e)

Set first derivative equal to zero to get the critical points

(0,0) will be our critical point.

(0, 0) will be a saddle point.