Question

Find all critical points of the following function. f(x,y) = xy - x - y What are the critical points? Select the correct choice below and, if necessary, fill in the answer box within your choice. O A. The critical point(s) is/are 7. (Type an ordered pair. Use a comma to separate answers as needed.) O B. There are no critical points.

Please solve this.
(Calc 3)

Answer 1

f(x,y) = ry - -44

$\frac{\partial f(x,y)}{\partial x} = \frac{\partial }{\partial x}(xy - x^4 - y^4) = y - 4x^3$

$\frac{\partial f(x,y)}{\partial y} = \frac{\partial }{\partial y}(xy - x^4 - y^4) = x - 4y^3$

Equating both these points to obtain the critical points

So, this will give the critical points as x=0 or x = -1/2,1/2

Hence the set of critical points are

(0,0), (1/2,1/2) and (-1/2,-1/2)

Imaginary number of pairs are (i/2,-i/2) and (-i/2,i/2)

Note - Post any doubts/queries in comments section.