Local extrems of two variable function I have a function: $f(x,y)=x^3 + y^3(8x^2-6xy+3y^2)$ 
I have problem with finding local extrems, more precisely I think that the function doesn't have any local extrem, but I have problem to prove it. I did a partial derivations and only point that could be extreme is $[x,y] = [0,0]$. Hessian matrix came out to be zero in that point. So what should i do? Thank you for response.

Since $f$ is differentiable, the only points where a local extreme value might occur are points where both partial derivatives evaluate to zero.

As you noted, there is only one such point, namely $(x,y)=(0,0)$.

But as $(x,y)$ approaches $(0,0)$ along the line $y=0$, we have
$$
f(x,y)=f(x,0)=x^3
$$
which yields positive outputs for $x > 0$, and negative outputs for $x < 0$.

Hence, since $f(0,0)=0$, it follows that $f$ does not have a local minimum or maximum at $(0,0)$.

Therefore $f$ does not have any local extrema.
