Critical points of 2-Variable Function

[Saladsamurai]

Homework Statement

Find local max/min, and saddle points (if any) of

$$f(x,y)=x^2+y^2+x^2y+4$$

This should be simple, but I am having algebra-block on solving the partial derivatives to find the critical points.

$f_x=2x+2xy=0$ (1)
$f_y=2y+x^2=0$ (2)

If I multiply the second equations by -x an add it to the second and solve for x, I get

x={0,+sqrt2, -sqrt2}

But for some reason I cannot figure out how to solve equation 2 ?

Why am I retarded?

 

[gabbagabbahey]

Well, for x=0 equation 2 gives: $2y+(0)^2=0 \Rightarrow y=0$ How about for x=+sqrt2?

 

[Saladsamurai]

I guess I don't understand this.

I want to know at what points the slope of the tangent lines is 0.

Now, I need values of x and y that satisfy both equations simultaneously. I am used to solving the equations simultaneously, not by assigning specific values to x or y.

I am not sure why that bothers me so much. But as you say:

if x=0 then eq 2 is satisfied by y=0, thus (0,0) is critical

if x=+ or - sqrt2 eq 2 is satisfied by y=-1, thus (+sqrt2, -1) and (-sqrt2, -1) are critical, yes?

 

[gabbagabbahey]

Yes, you can double check that those points satisfy both equations by substituting each point into them.