Solving 2 Problems: Finding Critical Points & Differentiability

[gassi]

I have two problems. I posted the first problem before but I still can´t solve it.

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.

I know this is symmetric but I cant´t figure out were to go from here??

Here is the second problem:

I have a function from R to R, f(x) = x + 2*x^2*sin(1/x) if x is not 0 and f(x) = 0 if x is 0.
I´m supposed to show that this is differentable everywhere, that f'(x) is not 0 and that f maps no neighbourhood around the zero point bijective on a neighbourhood around the zero point.

I think I know how to show that this is differentable everywhere, that f'(x) is not 0 but I´m having difficulties with the last one.

 

[gassi]

In the first can I subtracct df/dx - df/dy and get y - x - 3x^2 - 3y^2 = 0 which gives
y(1-3y) = x(1-3x). Therefore x=y and then get that x=z=y??

I also have to tell wether they are maximum, minimum or saddle points.