I have to determine the type of critical points of the function: 
$$f(x, y, z)=x^2+y^2+z^2+xy$$ 

I have done the following: 

$$\nabla f=(2x+y, 2y+x, 2z) \\ \nabla f=0 \Rightarrow \\ 2x+y=0 \Rightarrow y=-2x \\ 2y+x=0 \Rightarrow 2(-2x)+x=0 \Rightarrow -4x+x=0 \Rightarrow -3x=0 \Rightarrow x=0 \\ 2z=0 \Rightarrow z=0$$ 

So, the only critical point of $f$ is the point $(0, 0, 0)$. 

Is this correct so far?? 

How can I determine the type of the critical point, if it is a local maxima or a local minima?? 

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**EDIT:** 

$$f(x, y, z)=x^2+y^2+z^2+xy \\ =\frac{1}{2}x^2+xy+\frac{1}{2}y^2+z^2+\frac{1}{2}x^2+\frac{1}{2}y^2 \\ =\frac{1}{2}(x^2+2xy+y^2)+z^2+\frac{1}{2}(x^2+y^2) \\ =\frac{1}{2}(x+y)^2+z^2+\frac{1}{2}(x^2+y^2) \geq 0$$