Respuesta :
Answer:
Step-by-step explanation:
Given is a function f in two variables as
[tex]f(x, y) = (x − y)(xy − 9)[/tex]
To find critical points of the given function and also to find max, min or saddle point.
Find the partial derivatives
[tex]f(x,y) = x^2y-xy^2-9x+9y[/tex]
[tex]f_x=2xy-y^2-9\\f_y =x^2-2xy+9\\f_{xy} =2x-2y = f_{yx}[/tex]
[tex]f_{xx}=2y\\ f_{yy}=-2x[/tex]
Equate first derivatives to 0
Adding fx and fy we get
[tex]x^2-y^2 =0\\[/tex]
x=±y
Only real roots are (x,y) =(3,3) and (-3,-3)
[tex]D(3,3)=f_{xx}(3,3) f_{yy}(3,3)-f_{xy}^2(3,3) \\=-36-0<0[/tex]
(3,3) is a saddle point
[tex]D(-3,-3) = -36<0[/tex]
(-3,-3) is also a saddle point.
