Question

Find the 2 critical points of the given function and then determine whether it is a local maximum, local minimum, or saddle point. (Order your answers from smallest to largest x, then from smallest to largest y.)
f(x, y) = (x − y)(xy − 9)

Answer 1

Answer:

Step-by-step explanation:

Given is a function f in two variables as

$f(x, y) = (x − y)(xy − 9)$

To find critical points of the given function and also to find max, min or saddle point.

Find the partial derivatives

$f(x,y) = x^2y-xy^2-9x+9y$

$f_x=2xy-y^2-9\\f_y =x^2-2xy+9\\f_{xy} =2x-2y = f_{yx}$

$f_{xx}=2y\\ f_{yy}=-2x$

Equate first derivatives to 0

Adding fx and fy we get

$x^2-y^2 =0$

x=±y

Only real roots are (x,y) =(3,3) and (-3,-3)

$D(3,3)=f_{xx}(3,3) f_{yy}(3,3)-f_{xy}^2(3,3) \\=-36-0$

(3,3) is a saddle point

$D(-3,-3) = -36$

(-3,-3) is also a saddle point.