Question

How do you find the critical points of the function `f(x,y)=x^2+y^2+(x^2)(y)+4`?

Answer 1

(0,0), (`sqrt2`,-1) and (-`sqrt2`,-1)

Explanation:

For the given function, `f_x` = 2x +2xy and `f_y= 2y +x^2`. Equate `f_x`and `f_y` to 0 to get x(1+y)=0 and y= `-(1/2) x^2`. Solving these two

equations, we have `x(1-1/2 x^2)=0.` This gives x=0,`sqrt 2, -sqrt2` and

corresponding values of y =0,-1,-1.

The critical points are (0,0), (`sqrt 2`,-1) and (-`sqrt 2`,-1)