Classifying Critical Points of Multivariable Functions

[gassi]

Homework Statement

Find and classify the critical points of f(x,y,z) = xy + xz + yz + x^3 + y^3 + z^3

Homework Equations

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The Attempt at a Solution

df/dx = y + z +3x^2, df/dy = x + z + 3y^2 and df/dz = x + y + 3z^2

a point x is a critacal point if the gradient equals 0.

Obviously (0,0,0) is a critical point but I´m not sure how to find the others.

 

[tiny-tim]

Obviously (0,0,0) is a critical point but I´m not sure how to find the others.

Hi gassi! Welcome to PF!

Hint: the whole problem is completely symmetric in x y and z, isn't it?

So try adding or subtracting df/dx df/dy and df/dz.