Need help factoring in two variables

[Dannbr]

Homework Statement

Trying to find relative min/max

Homework Equations

f(x,y)=X^4+y^4-36xy

The Attempt at a Solution

partial WRT(x) = 4x^3-36y
partial WRT(y) = 4y^3-36x

Set Partial WRT(x) = 0

4x^3-36y=0
4x^3=36y
x=(9y)^1/3

plug x into Partial

4[(9y)^1/3]^3-36y=0
36y-36y=0

Im not sure what to do now to find the critical points?

 

[Harrisonized]

Let ∂f/∂x=0 and ∂f/∂y=0.

From your original post, you have

∂f/∂x = 4x³-36y = 4(x³-9y) = 0
∂f/∂y = 4y³-36x = 4(y³-9x) = 0

Now all you have left to do is to find all points that satisfy both equations. Two equations. Two unknowns. Algebra holds the answer.

 

[Dick]

Homework Statement

Trying to find relative min/max

Homework Equations

f(x,y)=X^4+y^4-36xy

The Attempt at a Solution

partial WRT(x) = 4x^3-36y
partial WRT(y) = 4y^3-36x

Set Partial WRT(x) = 0

4x^3-36y=0
4x^3=36y
x=(9y)^1/3

plug x into Partial

4[(9y)^1/3]^3-36y=0
36y-36y=0

Im not sure what to do now to find the critical points?

You just plugged your solution for WRT(x) back into the equation WRT(x). You aren't going to get anything out of that. Try solving the WRT(x) equation for y and putting that into the WRT(y) equation. You'll get an equation of high degree, but it's pretty easy to say what the possible real roots are.

 

[Dannbr]

Thanks for help