SUBJECT:Mathematical Economic

COURSE TEACHER:ANASHUA ANANGA MISS

PRESENTED BY:Rashmi Riztan MowriID NO:16141051

OPTIMIZATION OF

MULTIVARIABLE FUNCTIONS

Optimization Is The Process Of Finding The Relative Maximum Or Minimum Of a Function

For a Multivariable Function Such As Z=f(x,y) To Be at a Relative Minimum Or Maximum, three Conditions Must Be Met :

1. The first order partial derivatives must equal zero simultaneously.This indicates that at given point (a,b) called a critical point,the function is neither increasing nor decreasing with respect to the principal axes but is at a relative plateau.

2. The second order direct partial derivatives,when evaluated at the critical point (a,b) must both be negetive for a relative maximum and positive for a relative minimum. This ensures that from a relative plateau at (a,b) the function is concave and moving downward in relation to the principal axes in the case of a maximum and convex and moving upward in relation to the principal axes in the case of a minimum.

3. The product of the second order direct partial derivatives evaluated at the critical point must exceed the product of the cross partial derivatives also evaluated at the critical point. This added condition is needed to preclude an inflection point or saddle point.

In sum as seen in below figure, when evaluated at a critical point (a,b)

z

o

x

y

Zx=0

Zy=0

z

o

xy

Zx=0

Zy=0

Relative maximum: Relative minimum:

1.ƒx,ƒy=02.ƒxx,ƒyy>03.ƒxx,ƒyy>(ƒxy)

1.ƒx,ƒy=02.ƒxx,ƒyy<03.ƒxx,ƒyy>(ƒxy)

2 2

Note the following :1) since ƒxy = ƒyx by Young’s theorem, ƒxy.ƒyx = (ƒxy) . Step 3 can also be written ƒxx.ƒyy-(ƒxy) >0.

2) if ƒxx.ƒyy<(ƒxy) , when fxx and fyy have the same signs,the function is at an inflection point;

2

2

2

when fxx and fyy have different signs , the function is at a saddle point , as seen in below figure , where the function is at a maximum when viewed from one axis but at a minimum when viewed from the other axis.

z

x

y

3) if ƒxx.ƒyy = (ƒxy) , the test is inconclusive.

4) If the function is strictly concave(convex) in x and y , as in first figure, there will be only one maximum(minimum) , called an absolute or global maximum(minimum).

2

if the function is simply concave(convex) in x and y on an interval , the critical point is a relative or local maximum(minimum).

Lets do a math

(a) Find the critical points(b)Test whether the function is at a relative maximum or minimum given, z = 2y – x + 147x -54y + 12

3 3

a) Take the first order derivatives, sets them equal to zero and solve for x and y z = -3x +147 = 0 z = 6y – 54 = 0 x = 49 y = 9 x = +7 y = +3 with x = +7 , y = +3 , there are four distinct sets of critical points : (7,3) , (7,-3) , (-7,3) , (-7,-3)

2 2

2 2x y

- -- -

b) Take the second order direct partials from previous two equation evaluate them at each of the critical points and check the signs :

z = -6x1.z (7,3)= -6(7)

= -

42<0 2.z (7,-3)= -6(7)

= -

42<0 3.z (-7,3)= -6(-

7) =

42>04.z (-7,-3)=-6(-

7)

=42>0

z = 12y 1.z (7,3)= 12(3) = 36>0 2.z (7,-3)= 12(-3) = -36<0

3.z (-7,3)= 12(3) = 36>04.z (-7,-3)=12(-3) =-36<0

xx yy

xx

xx

xx

xx yy

yy

yy

yy

Since there are different signs for each of the second direct partials in (1) and (4),the function cannot be at a relative maximum or minimum at (7,3) or (-7,-3).when f and f are of different signs, f .f cannot be greater than (f ) , and the function is at a saddle point.

yyyy xx

xx

xy

2

With both signs of the second direct partials negative in (2) and positive in (3) ,the function may be at a relative maximum at (7,-3) and at a relative minimum at (-7,3) but the third condition must be tested first to ensure against the possibility of an inflection point.

c) Take the cross partial derivatives and check to make sure that z (a,b) . z (a,b)>[z (a,b)] z = 0 z = 0 z (a,b).z (a,b) >[z (a,b)]

xx

yy xy

2

2xy

xx yy

xy

yx

from (2), (-42) . (-36) > (0) 1512 > 0 from (3), (42) . (36) > (0) 1512 > 0

the function is maximized at (7,-3) and minimized at (-7,3) .

2

2

References : 1. GOOGLE2. INTODUCTION TO MATHEMATICAL ECONOMICS-EDWARD T.DOWLING , PH.D