Answer to Question #178017 in Microeconomics for Ratna

Question

Let Z = f(x,y) = 3x3-5y2-225x + 70y + 23.

(i)

Find the stationary points of z.

(ii)

Determine if at these points the function is at a relative maximum, relative 

minimum, infixion point, or saddle point.

Answer 1

ANSWER:

Let Z=f(x,y)=3x²-5y²-225x+70y+23

$\tfrac{\partial~f(x,y)}{\partial~x} =9x^{2} -225 =0$

x_o =$\sqrt{\dfrac{225}{9}}$ =5

$\tfrac{\partial~f(x,y)}{\partial~y} =-10 y +70 =0$

y=-7

D=$\begin{vmatrix} \tfrac{\partial{^2}~f(x_o,y_o)}{\partial~x^{2}} & \tfrac{\partial{^2}~f(x_o,y_o)}{\partial~y{\partial~x}} \\ \tfrac{\partial{^2}~f(x_o,y_o)}{\partial~x~\partial~y} & \tfrac{\partial{^2}~f(x_o,y_o)}{\partial~y^{2}} \end{vmatrix}$

D=$\begin{vmatrix} 90 & 0 \\ 0 & -10 \end{vmatrix}$

D=90*-10=-900

(i). Find the stationary points of z.

Point D is less than zero, therefore it is a stationary point.

(ii)Determine if at these points the function is at a relative maximum, relative 

minimum, infixion point, or saddle point.

(x_0,y₀) =(5,-7) is a saddle point.